fix(sessions): leave a pending idle alert to the user through a keep-alive turn

Only a human spends an idle alert, but the keep-alive turn resolved an unseen one three ways: its working transition, its stop hook, and its own idle prompt replacing the item with a pre-acknowledged one. While the turn runs, a pending idle item is now left exactly as found, the page's stop handler returns on the keep-alive reason before clearing hooks or raising Response Complete, and session:working/session:idle carry keepAlive so the stuck-detection timer neither restarts nor re-arms.

The window that marks the turn closes on its own idle prompt or any other submit instead of a fixed three minutes, which a slow turn's idle prompt (sent a minute after the turn ends) could outlast. The service keeps the default and cap in memory, read through a strict reader so a mid-write read is retried rather than kept, and the cap is clamped before the strict settings save.

docs/wiki/Keeping-Agents-Running.md

@@ -113,9 +113,9 @@ Claude Code 2.1.251 and newer reports the cache state on the statusline after ev
 
 It only fires when there is nothing else going on: the session is idle with an empty prompt, no permission or question dialog is open, nobody has typed in the last few minutes, no respawn loop, Ralph loop or waiting agent is driving the session, and the session is not paused on a usage limit. If any of those hold it waits a minute and looks again, and gives up a minute before the expiry rather than race it. It also stops by itself a few hours after your last prompt (**Keep warm for at most**, under App Settings → System → Automation, default 2 hours), because the saving is concentrated in gaps of one to four hours and an overnight run of hourly keep-alives would cost more than the single cold turn in the morning. It stands down for the rest of a gap once the context has been compacted. The readout's Status line gains a `Keep-alive:` line that says what it is doing: the next keep-alive and when the window closes, waiting and why, sent at a time with a count, or stopped and why. A session started without the cache readout (Plan Usage off at launch, a workspace with its own statusLine, a remote or Docker session) cannot be kept warm: the switch is greyed out and cannot be switched on (one that is already on can still be switched off), and the Status line says why.
 
-Recent Claude Code builds can compact a large idle session on their own shortly before the cache expires, which would leave only a summary to keep warm. So a session **started** with keep-warm on also launches with Claude Code's idle compaction switched off (Claude Code 2.1.290 or newer); Codeman then owns the idle policy for it. A session you switch on after it started is still kept warm, but Claude Code may compact it first until its next launch, and the Status line says so. Sessions with keep-warm off launch exactly as before.
+Recent Claude Code builds can compact a large idle session on their own shortly before the cache expires, which would leave only a summary to keep warm. So a session **started** with keep-warm on also launches with Claude Code's idle compaction switched off (Claude Code 2.1.290 or newer); Codeman then owns the idle policy for it. A session you switch on after it started is still kept warm, but Claude Code may compact it first until its next launch, and the Status line says so; one you switch off keeps Claude Code's idle compaction off until its next launch, and the Status line says that too. Sessions with keep-warm off launch exactly as before.
 
-**Keep prompt caches warm** in the same Automation group is the default for sessions that have not chosen on their Respawn tab; the per-session switch always wins. A session another agent started (a worker) does not follow the default: most workers are never prompted again once they report done, so turn it on for one by hand if you will come back to it. The keep-alive turn is Codeman's, not yours: it does not restart the **Keep warm for at most** window, does not rename a tab, Read My Mind ignores it, and the response it ends with raises no alert, toast, push or webhook. It is refused on the five-minute cache lifetime, which is what an API key, a cloud provider or extra-usage credits get; a session on an API key can ask Claude Code for the hour with `CLAUDE_CODE_PROMPT_CACHE_TTL=1h` in its environment overrides.
+**Keep prompt caches warm** in the same Automation group is the default for sessions that have not chosen on their Respawn tab; the per-session switch always wins. A session another agent started (a worker) does not follow the default: most workers are never prompted again once they report done, so turn it on for one by hand if you will come back to it. The keep-alive turn is Codeman's, not yours: it does not restart the **Keep warm for at most** window, does not rename a tab, Read My Mind ignores it, and the response it ends with raises no alert, toast, push or webhook. It never clears an alert that was already waiting for you. It is refused on the five-minute cache lifetime, which is what an API key, a cloud provider or extra-usage credits get; a session on an API key can ask Claude Code for the hour with `CLAUDE_CODE_PROMPT_CACHE_TTL=1h` in its environment overrides.
 
 ## The plan usage chip