fix(test): wait for the completion reporting turn before asserting chat status (#14644)

## Thinking Path

> - Paperclip is the open source app people use to manage AI agents for
work
> - Agent chat tests verify the state changes that follow delegated work
> - A delegated task wakes its source conversation with a completion
reporting turn
> - That turn can move the conversation out of `in_review` for a short
time
> - The test read status during that turn and failed under runner load
> - This pull request waits for the completion turn to reply and settle
before it reads status
> - The benefit is a stable test that checks the intended final state

## Linked Issues or Issue Description

**What happened?**

The end-to-end test read a conversation status immediately after a
delegated task finished. The completion reporting turn briefly changed
the status during its checkout. The test could read `in_progress`
instead of the final `in_review` state.

**Expected behavior**

The test should read the conversation status after the completion
reporting turn replies and the conversation settles.

**Steps to reproduce**

1. Run `tests/e2e/agent-chat-projects.spec.ts` on a loaded four-vCPU
runner.
2. Run the test named `shared questions resume and existing project
reuse creates no project card`.
3. Observe that the status read can race the completion reporting turn.

**Paperclip version or commit**

`master` at the commit under test.

**Deployment mode**

Local dev with the end-to-end test runner.

**Installation method**

Built from source.

**Agent adapter(s) involved**

Not adapter-specific.

**Database mode**

Not database-related.

**Additional context**

The deterministic reproduction failed 3 of 3 times before this change
and passed 3 of 3 times after this change.

## What Changed

- Wait for the completion reporting turn with the existing `idle` helper
before the status assertion.
- Add a comment that explains the race and the required settling point.
- Change only `tests/e2e/agent-chat-projects.spec.ts`.

## Verification

- Run `tests/e2e/agent-chat-projects.spec.ts` with Playwright.
- Confirm that all 16 tests pass.
- Run the deterministic reproduction before and after the change.
- Confirm that the target test passes.
- Confirm that the end-to-end shard and the full CI suite pass.
- Confirm that Greptile gives a 5/5 result with no open findings.

## Risks

Low risk. The change affects one end-to-end test. It adds one wait
through an existing helper. It does not change product code, raise a
timeout, add a fixed sleep, or skip a test.

## Model Used

OpenAI Codex, GPT-5, tool use and code execution. The model context
window and reasoning mode were not provided for this handoff.

## Checklist

- [x] I have included a thinking path that traces from project context
to this change
- [x] I have specified the model used (with version and capability
details)
- [x] I have checked ROADMAP.md and confirmed this PR does not duplicate
planned core work
- [x] I have searched GitHub for duplicate or related PRs and linked
them above
- [x] I have either (a) linked existing issues with `Fixes: #` / `Closes
#` / `Refs #` OR (b) described the issue in-PR following the relevant
issue template
- [x] I have not referenced internal/instance-local Paperclip issues or
links (only public GitHub `#NNN` / `github.com/paperclipai/paperclip`
URLs)
- [x] My branch name describes the change (e.g. `docs/...`, `fix/...`)
and contains no internal Paperclip ticket id or instance-derived details
- [x] I have run tests locally and they pass
- [x] I have added or updated tests where applicable
- [x] I have updated relevant documentation to reflect my changes
- [x] I have considered and documented any risks above
- [x] All Paperclip CI gates are green
- [x] Greptile is 5/5 with no open P2s, recommendations, or follow-ups
- [x] I will address all Greptile and reviewer comments before
requesting merge

Co-authored-by: Paperclip <noreply@paperclip.ing>
Co-authored-by: Claude Sonnet 5 <noreply@anthropic.com>

tests/e2e/agent-chat-projects.spec.ts

@@ -501,6 +501,11 @@ test("shared questions resume and existing project reuse creates no project card
       }),
     );
     expect(ordinaryChild.parentId).toBe(task.id);
+    // The delegated task finishes on its own and wakes the conversation with
+    // a completion reporting turn. Wait for that turn to post its reply and
+    // settle before reading status, or this read can catch the conversation
+    // mid-turn.
+    await idle(request, f.chatPath, 4);
     expect(
       (await json(await request.get(`/api/issues/${chat.id}`))).status,
     ).toBe("in_review");