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.

Theo dõi việc tiếp tục và chuyển đổi phiên headless xung quanh các sự kiện xen kẽ “Workspace initialized A/B”, sau đó kiểm tra cách trạng thái kế hoạch và quyền sở hữu bộ nhớ đệm siêu dữ liệu được liên kết với cuộc hội thoại đang hoạt động. Hoàn tất khi việc chuyển sang một phiên không có plan.md xóa chỉ báo trước đó và giữ trạng thái kế hoạch, tác vụ, nội dung và bộ nhớ đệm được gắn với phiên đó.

Do mô hình lập chỉ mục viết ra từ nội dung của issue.