Desktop: "Suggested task" chip card always defaults to "Start with worktree" - saved spawn-target preference regressed (~Jul 21)

### Describe the bug

Since ~2026-07-21 the "Suggested task" popup card in Claude Code Desktop (background task suggestions created via the `spawn_task` tool) always shows **"Start with worktree"** as the primary action of its split button. "Start locally" only exists in the dropdown, and the choice is not remembered - the next chip is back to "Start with worktree".

Until ~Jul 20 the primary action reflected the persisted target preference (`epitaxy-spawn-task-target-v2`, values local/worktree/cloud/here, menu items with checkmarks). Users who had chosen "Start locally" got local starts by default. That preference is still stored in `claude_desktop_config.json` (`preferences.epitaxyPrefs`) but the card no longer reads it, and a selection made in the dropdown is no longer persisted anywhere.

### Steps to reproduce

1. Have any session create a task suggestion via the `spawn_task` tool.
2. On the chip card, open the dropdown next to "Start with worktree" and pick "Start locally".
3. Let the next chip appear.
4. The primary action is "Start with worktree" again - the explicit choice was not persisted (reproduced 10+ times).

### Expected behavior

Remember the last explicitly chosen start target (the pre-Jul-21 behavior), or offer a setting to disable defaulting to worktrees for local sessions.

### Environment

- Claude Desktop 1.24012.9 (Windows 11 Pro)
- Web frontend build 2026-07-25
- Bundled Claude Code CLI 2.1.219

Start at the Desktop "Suggested task" chip card and its spawn_task flow, then inspect claude_desktop_config.json and preferences.epitaxyPrefs for the epitaxy-spawn-task-target-v2 value. Trace how the split-button selection is read and saved, and verify that a later chip uses the last explicitly chosen target, including "Start locally".

Written by the indexing model from the issue text.