ci: Pull out in-browser tests into their own parallelized job

.github/workflows/test_web.yml

@@ -93,13 +93,55 @@ jobs:
         working-directory: web
         run: npm test
 
+      - name: Upload artifacts
+        if: matrix.os == 'ubuntu-22.04' && matrix.node_version == '22'
+        uses: actions/upload-artifact@v4
+        with:
+          name: selfhosted-dist
+          path: web/packages/selfhosted/dist
+
+  browser-tests:
+    # Only the instance that uploads the artifact is really needed, but there's no way to express that.
+    # See: https://github.com/orgs/community/discussions/42335
+    needs: build
+    if: needs.changes.outputs.should_run == 'true'
+    name: Test in ${{ matrix.browser }}
+    runs-on: ubuntu-22.04
+    strategy:
+      fail-fast: false
+      matrix:
+        browser: ["chrome", "firefox", "edge"]
+
+    steps:
+      - uses: actions/checkout@v4
+
+      - name: Setup Node.js 22
+        uses: actions/setup-node@v4
+        with:
+          node-version: 22
+          cache: npm
+          cache-dependency-path: web/package-lock.json
+
+      - name: Download artifacts
+        uses: actions/download-artifact@v4
+        with:
+          name: selfhosted-dist
+          path: web/packages/selfhosted/dist
+
       - name: Run browser-based tests
-        # Don't run browser tests on Windows because it's flaky for unknown reasons. :-(
-        if: runner.os == 'Linux'
         working-directory: web
-        run: npm run wdio -- --headless --chrome --firefox --edge
+        shell: bash -l {0}
+        run: |
+          npm ci
+          npm run wdio -- --headless --${{ matrix.browser }}
 
-  check-required:
+  # The rest is needed only because otherwise the real jobs couldn't be made required for merging PRs,
+  # as they would fail the entire workflow run if skipped - this makes GitHub happy instead.
+  # See:
+  #  - https://github.com/orgs/community/discussions/13690
+  #  - https://github.com/orgs/community/discussions/44490
+
+  build-dummy:
     needs: changes
     if: needs.changes.outputs.should_run == 'false'
     name: Test Node.js ${{ matrix.node_version }} / ${{ matrix.os }}
@@ -112,3 +154,16 @@ jobs:
     steps:
       - name: No-op
         run: echo noop
+
+  browser-tests-dummy:
+    needs: build
+    if: needs.changes.outputs.should_run == 'false'
+    name: Test in ${{ matrix.browser }}
+    runs-on: ubuntu-22.04
+    strategy:
+      matrix:
+        browser: ["chrome", "firefox", "edge"]
+
+    steps:
+      - name: No-op
+        run: echo noop