Child writing tasks can retain plan-mode write gates after approved exit

### Describe the bug

A background writing task launched after an approved plan-mode exit can retain a stale plan-mode write gate. This falsely blocks the task or applies different gates to its tools and siblings, consuming bounded retry budget and potentially stalling fleet execution.

In GitHub Copilot CLI 1.0.71, after the root received `Plan approved! Exited plan mode.` and entered fleet/autopilot execution:

- One writing task launched after the approved exit created its isolated worktree and read sources, but its first harmless `apply_patch` was rejected with the inherited-parent-plan denial.
- A replacement launched after another explicit exit still had write-capable bash commands rejected as plan-mode writes, although its edit tool and commit worked.
- Sibling writing tasks launched during the same fleet phase edited and committed successfully.

### Affected version

GitHub Copilot CLI 1.0.71

### Steps to reproduce the behavior

1. Start a fresh GitHub Copilot app-managed project session.
2. Enter plan mode and prepare a plan.
3. Approve the exit to autopilot/fleet and wait for the root to report `Plan approved! Exited plan mode.`.
4. Immediately launch one or two native background general-purpose writing tasks.
5. Have each task make a first harmless write.
6. Observe whether a child receives the inherited-parent-plan denial, whether different write-capable tools in one child are gated differently, or whether sibling children receive different mode state.

### Expected behavior

After the approved plan exit and entry into fleet/autopilot execution, every newly launched writing task's plan-mode gate is inactive. Child gates should track the parent's live mode state, consistent with the existing runtime test requirement, rather than a stale state captured before exit.

### Additional context

- **Environment:** GitHub Copilot CLI 1.0.71 in a GitHub Copilot app-managed project session using native background general-purpose tasks.
- **Likely component, not yet confirmed:** mode propagation around `PlanModeWriteGateProcessor` in `github/copilot-agent-runtime`. The isolated worktree appears incidental because sibling worktrees succeeded during the same fleet phase.
- **Impact:** A task can be falsely blocked or inconsistently tool-gated, consume the fleet's bounded retry budget, and stall otherwise independent fleet work.

Related open reports cover plan-approval fleet activation or ordering, but not a child retaining a stale write gate after the approved exit:

- https://github.com/github/copilot-cli/issues/1901
- https://github.com/github/copilot-cli/issues/2833

#### Acceptance criteria

- [ ] A child launched after an approved plan exit receives the parent's live non-plan state.
- [ ] Runtime regression coverage exercises a background child launched after the exit.
- [ ] Task/fleet regression coverage exercises the sibling-launch race after the exit.
- [ ] Write-capable tools within a child consistently enforce the same current mode state.

Start by tracing mode propagation around PlanModeWriteGateProcessor in github/copilot-agent-runtime and review the existing runtime test requirement for child mode state. Add regression coverage for a background child launched after an approved plan exit and for the sibling-launch race in task or fleet tests. Done means children and their write-capable tools consistently use the parent's live non-plan state.

Written by the indexing model from the issue text.