diff --git a/Polygon_mesh_processing/include/CGAL/Polygon_mesh_processing/internal/Corefinement/Face_graph_output_builder.h b/Polygon_mesh_processing/include/CGAL/Polygon_mesh_processing/internal/Corefinement/Face_graph_output_builder.h
index e4d9f60d1a2..44c4756c9ac 100644
--- a/Polygon_mesh_processing/include/CGAL/Polygon_mesh_processing/internal/Corefinement/Face_graph_output_builder.h
+++ b/Polygon_mesh_processing/include/CGAL/Polygon_mesh_processing/internal/Corefinement/Face_graph_output_builder.h
@@ -1091,6 +1091,10 @@ class Face_graph_output_builder
     boost::dynamic_bitset<> coplanar_patches_of_tm2_for_union_and_intersection(nb_patches_tm2,false);
     patch_status_not_set_tm1.set();
     patch_status_not_set_tm2.set();
+    // extra containers used when entire connected components are identical (filled only if needed)
+    std::vector<std::size_t> coplanar_tm1_to_coplanar_tm2;
+    std::vector<vertex_descriptor> extreme_vertex_per_cc_1;
+    std::vector<vertex_descriptor> extreme_vertex_per_cc_2;
 
     // first set coplanar status of patches using the coplanar faces collected during the
     // extra intersection edges collected. This is important in the case of full connected components
@@ -1107,71 +1111,6 @@ class Face_graph_output_builder
       coplanar_patches_of_tm2.set(tm2_patch_ids[ fid2 ]);
     }
 
-
-    //we have the correspondance between cpln patches thanks to faces in tm1_coplanar_faces and tm2_coplanar_faces
-    CGAL_assertion(tm1_coplanar_faces.size()==tm2_coplanar_faces.size());
-    // TODO: move it to do it lazily
-    std::vector<std::size_t> coplanar_tm1_to_coplanar_tm2;
-    std::vector<bool> is_oo_tm1(nb_patches_tm1, false);
-    std::vector<bool> is_oo_tm2(nb_patches_tm2, false);
-    if (!tm1_coplanar_faces.empty())
-    {
-      coplanar_tm1_to_coplanar_tm2.resize(nb_patches_tm1, NID);
-      for (std::size_t i=0; i<tm1_coplanar_faces.size(); ++i)
-      {
-        std::size_t pid1 = tm1_patch_ids[get(fids1, tm1_coplanar_faces[i])];
-        std::size_t pid2 = tm2_patch_ids[get(fids2, tm2_coplanar_faces[i])];
-        coplanar_tm1_to_coplanar_tm2[pid1]=pid2;
-      }
-
-      std::vector<std::optional<vertex_descriptor>> max_pts_1(nb_patches_tm1);
-      for (face_descriptor fd : faces(tm1))
-      {
-        std::size_t patch_id = tm1_patch_ids[get(fids1, fd)];
-        if (!coplanar_patches_of_tm1.test(patch_id)) continue;
-        halfedge_descriptor hd=halfedge(fd, tm1);
-        for (halfedge_descriptor h : CGAL::halfedges_around_face(hd, tm1))
-        {
-          vertex_descriptor vd = target(h, tm1);
-          if (!max_pts_1[patch_id].has_value() || get(vpm1,max_pts_1[patch_id].value())<get(vpm1,vd))
-            max_pts_1[patch_id]=vd;
-        }
-      }
-
-      for (std::size_t pid=0;pid<nb_patches_tm1; ++pid)
-      {
-        if (coplanar_patches_of_tm1.test(pid)) // TODO restrict more!
-        {
-          is_oo_tm1[pid] = ::CGAL::Polygon_mesh_processing::internal::is_outward_oriented(max_pts_1[pid].value(), tm1, parameters::vertex_point_map(vpm1));
-        }
-      }
-
-
-      std::vector<std::optional<vertex_descriptor>> max_pts_2(nb_patches_tm2);
-      for (face_descriptor fd : faces(tm2))
-      {
-        std::size_t patch_id = tm2_patch_ids[get(fids2, fd)];
-        if (!coplanar_patches_of_tm2.test(patch_id)) continue;
-        halfedge_descriptor hd=halfedge(fd, tm2);
-        for (halfedge_descriptor h : CGAL::halfedges_around_face(hd, tm2))
-        {
-          vertex_descriptor vd = target(h, tm2);
-          if (!max_pts_2[patch_id].has_value() || get(vpm2,max_pts_2[patch_id].value())<get(vpm2,vd))
-            max_pts_2[patch_id]=vd;
-        }
-      }
-
-      for (std::size_t pid=0;pid<nb_patches_tm2; ++pid)
-      {
-        if (coplanar_patches_of_tm2.test(pid))
-        {
-          is_oo_tm2[pid] = ::CGAL::Polygon_mesh_processing::internal::is_outward_oriented(max_pts_2[pid].value(), tm2, parameters::vertex_point_map(vpm2));
-        }
-      }
-    }
-
-
-
     for (typename An_edge_per_polyline_map::iterator
             it=an_edge_per_polyline.begin(),
             it_end=an_edge_per_polyline.end(); it!=it_end;++it)
@@ -1912,10 +1851,56 @@ class Face_graph_output_builder
           {
             if (coplanar_patches_of_tm1.test(patch_id))
             {
+              // Two "identical" coplanar patches that are entire connected components
+              //we have the correspondance between cpln patches thanks to faces in tm1_coplanar_faces and tm2_coplanar_faces
+              CGAL_assertion(tm1_coplanar_faces.size()==tm2_coplanar_faces.size());
+              if (coplanar_tm1_to_coplanar_tm2.empty()) // fill container only once
+              {
+                coplanar_tm1_to_coplanar_tm2.resize(nb_patches_tm1, NID);
+                for (std::size_t i=0; i<tm1_coplanar_faces.size(); ++i)
+                {
+                  std::size_t pid1 = tm1_patch_ids[get(fids1, tm1_coplanar_faces[i])];
+                  std::size_t pid2 = tm2_patch_ids[get(fids2, tm2_coplanar_faces[i])];
+                  coplanar_tm1_to_coplanar_tm2[pid1]=pid2;
+                }
+
+                const vertex_descriptor null_v = boost::graph_traits<TriangleMesh>::null_vertex();
+                extreme_vertex_per_cc_1.assign(nb_patches_tm1, null_v);
+                for (face_descriptor fd : faces(tm1))
+                {
+                  std::size_t patch_id = tm1_patch_ids[get(fids1, fd)];
+                  if (!coplanar_patches_of_tm1.test(patch_id)) continue;
+                  halfedge_descriptor hd=halfedge(fd, tm1);
+                  for (halfedge_descriptor h : CGAL::halfedges_around_face(hd, tm1))
+                  {
+                    vertex_descriptor vd = target(h, tm1);
+                    if (extreme_vertex_per_cc_1[patch_id]==null_v || get(vpm1,extreme_vertex_per_cc_1[patch_id])<get(vpm1,vd))
+                      extreme_vertex_per_cc_1[patch_id]=vd;
+                  }
+                }
+
+                extreme_vertex_per_cc_2.assign(nb_patches_tm2, null_v);
+                for (face_descriptor fd : faces(tm2))
+                {
+                  std::size_t patch_id = tm2_patch_ids[get(fids2, fd)];
+                  if (!coplanar_patches_of_tm2.test(patch_id)) continue;
+                  halfedge_descriptor hd=halfedge(fd, tm2);
+                  for (halfedge_descriptor h : CGAL::halfedges_around_face(hd, tm2))
+                  {
+                    vertex_descriptor vd = target(h, tm2);
+                    if (extreme_vertex_per_cc_2[patch_id]==null_v || get(vpm2,extreme_vertex_per_cc_2[patch_id])<get(vpm2,vd))
+                      extreme_vertex_per_cc_2[patch_id]=vd;
+                  }
+                }
+              }
+
               const std::size_t patch_id2=coplanar_tm1_to_coplanar_tm2[patch_id];
               CGAL_assertion(patch_id2!=NID);
 
-              if (is_oo_tm1[patch_id]==is_oo_tm2[patch_id2])
+              bool is_oo_tm1 = ::CGAL::Polygon_mesh_processing::internal::is_outward_oriented(extreme_vertex_per_cc_1[patch_id], tm1, parameters::vertex_point_map(vpm1)),
+                   is_oo_tm2 = ::CGAL::Polygon_mesh_processing::internal::is_outward_oriented(extreme_vertex_per_cc_2[patch_id2], tm2, parameters::vertex_point_map(vpm2));
+
+              if (is_oo_tm1==is_oo_tm2)
               {
                 coplanar_patches_of_tm1_for_union_and_intersection.set(patch_id);
                 coplanar_patches_of_tm2_for_union_and_intersection.set(patch_id2);
diff --git a/Polygon_mesh_processing/test/Polygon_mesh_processing/test_corefinement_cavities.cpp b/Polygon_mesh_processing/test/Polygon_mesh_processing/test_corefinement_cavities.cpp
index 99719ed0cfb..ad5bfe39675 100644
--- a/Polygon_mesh_processing/test/Polygon_mesh_processing/test_corefinement_cavities.cpp
+++ b/Polygon_mesh_processing/test/Polygon_mesh_processing/test_corefinement_cavities.cpp
@@ -43,6 +43,9 @@ void test_operations(Mesh A, Mesh B,
                      std::string round,
                      std::size_t union_v, std::size_t inter_v, std::size_t diff1_v, std::size_t diff2_v)
 {
+#ifndef VERBOSE
+  CGAL_USE(round);
+#endif
   if (reverse_A) PMP::reverse_face_orientations(A);
   if (reverse_B) PMP::reverse_face_orientations(B);