[BUG] Desktop: per-session banner says background-task session will auto-archive, contradicting "Archive inactive sessions: Never"

## Environment
- Claude desktop app, Code tab
- Windows 11 Enterprise, build 10.0.26200

## Summary
A session running a background task shows a banner claiming it will be auto-archived once the background task finishes — even though the "Archive inactive sessions" setting is set to **Never**, and that setting's own description explicitly says sessions with running/background work are exempt.

## Repro
1. Set **Settings → Archive inactive sessions** to **Never**. The setting's full label and description read, verbatim:
> **Archive inactive sessions**
> Automatically archive local sessions after a period of no activity. Sessions that are running or have background work are never archived, and a worktree with uncommitted changes is kept on disk.

The dropdown next to it is set to **Never**.
2. Start a task that continues running in the background (in this case, asking Claude to verify a DNS fix, which kept running server-side after the chat turn ended).
3. The session shows this banner, verbatim:
> Running in the background — I'll be notified when it completes, then verify the actual DNS fix live.
>
> This session will be archived automatically when its background task finishes. **[Stop task and archive]**

("Stop task and archive" is a clickable link in the banner, not just text.)

## Expected
Per the "Archive inactive sessions" setting (Never, with background-work sessions explicitly stated as exempt), the session should **not** be scheduled for auto-archive, and the banner should not tell the user it will be.

## Actual
The banner unconditionally states the session will be archived once its background task completes, regardless of the "Archive inactive sessions" setting. This is either:
- a UI copy bug (the banner doesn't check the user's setting before rendering that line), or
- a genuinely separate auto-archive-on-background-task-completion mechanism that isn't gated by "Archive inactive sessions" at all, contradicting that setting's own description.

Either way there's a real discrepancy between what the settings page promises and what the in-session banner tells the user will happen. It isn't yet confirmed which side is actually wrong — that would need someone to let a background task finish while the setting is Never and check whether the session actually gets archived.

## Related issues
- #75548 (closed) — auto-archive killing actively running sessions despite background work
- #60043 — long working sessions auto-archive mid-conversation, no opt-out
- #75941 — sessions auto-archived repeatedly without user action

None of these document this specific setting-vs-banner contradiction.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Reproduce this in the Claude desktop app's Code tab on Windows 11 with Archive inactive sessions set to Never. Let a background task finish, then check whether the session is archived and whether the banner's auto-archive statement matches the setting; done means the setting and banner agree.

Written by the indexing model from the issue text.