fix(clerk-js): Avoid client updates on sign out #12887

Job	Run time
check-permissions	1s
Formatting, linting & changeset checks	2m 0s
Unit Tests (18)	50s
Unit Tests (22)	1m 25s
Integration Tests (generic, chrome)	4m 5s
Integration Tests (express, chrome)	2m 8s
Integration Tests (quickstart, chrome)	2m 56s
Integration Tests (ap-flows, chrome)	2m 40s
Integration Tests (elements, chrome)	2m 54s
Integration Tests (sessions, chrome)	3m 47s
Integration Tests (astro, chrome)	3m 1s
Integration Tests (expo-web, chrome)	2m 28s
Integration Tests (tanstack-start, chrome)	2m 11s
Integration Tests (tanstack-router, chrome)	2m 12s
Integration Tests (vue, chrome)	2m 4s
Integration Tests (nuxt, chrome)	2m 31s
Integration Tests (react-router, chrome)	2m 14s
Integration Tests (nextjs, chrome, 13)	4m 4s
Integration Tests (nextjs, chrome, 14)	4m 4s
Integration Tests (nextjs, chrome, 15)	4m 2s
	51m 37s

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