Refresh/keep session in sync with external modifications (e.g. via VS Code)

### Feature summary

_No response_

### What problem are you trying to solve?

After implementing something in the app, it's common to e.g. open VS Code to tweak/review/continue manual work. The Github App session is visible inside VS Code, and can be continued there, but the changes aren't picked up back in the Github App; the latter must be restarted for it to reread the session continuation done outside in VS Code.

### Proposed solution

In an ideal world, the App would simply detect external session and show them; since that's probably complicated (two UIs working on the same copilot session), at least some way to refresh the session , similar to what a restart does would help.

### Workflow impact

_No response_

### Installation context

_No response_

### Additional context

Note slight similarity with #1205, though that issue is more about picking up **new** sessions, whereas here it's about picking up continuations of existing/known sessions.

The issue names no files, tests, or entry points. Start by tracing how the GitHub App reads Copilot sessions and how continuations made in VS Code become visible after an app restart; done means an existing session can be refreshed to show external changes without restarting the app.

Written by the indexing model from the issue text.