Plan indicator leaks across conversations after headless session switches

### Describe the bug

When the IDE switches one headless Copilot CLI process between two conversations that share the same repository, worktree, and branch, the plan indicator/path can remain bound to the previously active conversation.

Observed result:

- Conversation A had a physical per-session `plan.md`.
- Conversation B had no `plan.md` and an empty per-session task database.
- While B was active, the IDE displayed A's session-specific `...\A\plan.md` beside B.
- The UI for A simultaneously claimed that A did not have a plan.
- Nothing had moved on disk: A's plan remained under A's session-state directory and B's directory had no plan.

The CLI/backend session state was correctly isolated; the visible conversation-to-plan association was reversed/stale.

### Affected version

```text
Copilot IDE 1.12.1
Copilot CLI backend 1.0.71
VS Code 1.130.0
Windows x64
```

### Steps to reproduce the behavior

1. Open two IDE conversations for the same repository/worktree/branch.
2. In conversation A, enter plan mode and create/update `plan.md`.
3. Leave conversation B without a plan.
4. Switch repeatedly between A and B so the IDE resumes both through the same headless CLI process.
5. Observe the plan indicator/path after B becomes active.

In the affected run, sanitized debug logs showed the same process repeatedly alternating:

```text
resume A -> Workspace initialized A
resume B -> Workspace initialized B
resume A -> Workspace initialized A
resume B -> Workspace initialized B
```

After the final `Workspace initialized B`, B displayed A's plan path. The local IDE metadata cache contained A but had no entry for B.

### Expected behavior

Plan state must be keyed and rendered strictly by conversation/session ID:

- Activating B must clear A's plan indicator before B is rendered.
- A conversation without `plan.md` must never inherit a plan based on shared cwd, repository, branch, process, or prior active state.
- Session resume/switch should atomically update conversation content, plan state, task state, and metadata-cache ownership.
- If the active session lacks a cache entry, the UI should query that session or show no plan rather than retaining the previous session's plan.

### Additional context

The backend evidence rules out a filesystem move or shared SQL state: each conversation retained its own session directory/database, and the active session's injected context pointed at the correct session folder. This appears to be stale IDE/session-controller state during rapid headless session switching.

The IDE package points to a separate private repository that is not accessible to either configured GitHub identity, so this is filed against the CLI/session tracker for routing. No private logs or conversation content are attached.

Trace headless session resume and switch handling around the alternating “Workspace initialized A/B” events, then inspect how plan state and metadata-cache ownership are associated with the active conversation. Done means switching to a session without plan.md clears the prior indicator and keeps plan, task, content, and cache state keyed to that session.

Written by the indexing model from the issue text.