[BUG] Desktop app: side pane arrangement (side by side vs stacked) resets after switching chats

In the Code tab, with two side panes open (the built-in browser and a plugin pane opened with $.ui.open), I arrange them side by side. After clicking another chat in the sidebar and coming back, both panes are still open, but they're stacked vertically (browser on top, plugin pane below) instead of side by side.

Steps

1. Open a session and open the built-in browser pane.
2. Open a plugin pane (e.g. a mod calling $.ui.open({ id, title })).
3. Arrange them side by side.
4. Click another chat in the sidebar, then return to the first one.

Expected: the panes keep the arrangement they had.
Actual: they come back stacked in the default arrangement.

The plugin API has no way to request a placement relative to other panes (PaneOpenArgs only has rows/columns), so a plugin can't work around this.

Related: #98514 (default layout for new sessions; this one is about an existing session losing its layout).

Environment: Claude desktop app (Code tab), macOS (Darwin 27.0.0), Claude Code 2.1.295.