Link to the code that reproduces this issue

https://github.com/cjonesdoordash/nextjs-prefetch-issue/tree/template-issue

To Reproduce

With next/link pre-fetching disabled (default when you pull down the branch)

1. Run yarn build && yarn start
2. Navigate to localhost:3000/testing - verify that you have two templates rendered with data from a template server component and state from a template client component (one outer and one inner)
3. Increment client-side counters to change template state on both counters
4. Navigate to the next page via the link
5. Verify that for the outer template neither the server seconds nor the client counter change
6. Verify for the inner template just the client counter resets

With next/link pre-fetching enabled (just remove the prefetch={false} on both links)

1. Run yarn build && yarn start
2. Navigate to localhost:3000/testing - verify that you have two templates rendered with data from a template server component and state from a template client component (one outer and one inner)
3. Increment client-side counters to change template state on both counters
4. Navigate to the next page via the link
5. Verify that for the outer template the client counter doesn't change, but the server seconds sometimes doesn't reflect a proper change in seconds.
6. Verify for the inner template that both the seconds increment correctly and the client counter resets
7. Using quick navigations back and forth to trigger the RSC client-side cache verify that the inner template seconds doesn't change once you hit the cache

Current vs. Expected behavior

Expected behavior is that Templates re-mount and render in their entirety (both server and client components) on any route transition if they are within the segment.

Current behavior is that Templates fail to mount / render properly in a bunch of difference scenarios listed about in the reproductions.

Verify canary release

• I verified that the issue exists in the latest Next.js canary release

Provide environment information

Operating System:
  Platform: darwin
  Arch: x64
  Version: Darwin Kernel Version 22.6.0: Fri Sep 15 13:39:52 PDT 2023; root:xnu-8796.141.3.700.8~1/RELEASE_X86_64
Binaries:
  Node: 16.18.0
  npm: 8.19.2
  Yarn: 1.22.5
  pnpm: 6.23.6
Relevant Packages:
  next: 13.5.4
  eslint-config-next: N/A
  react: 18.2.0
  react-dom: 18.2.0
  typescript: 5.2.2
Next.js Config:
  output: N/A

Which area(s) are affected? (Select all that apply)

App Router, Routing (next/router, next/navigation, next/link)

Additional context

No response