[FEATURE] Desktop: restore the split-view layout on launch, plus saved layouts or workspace tabs

Preflight Checklist

• [x] I have searched existing requests. #88020 (named split-view layouts) and #86925 (split view as the default) asked for this and were auto-closed for inactivity, not declined. The bot said to open a new issue if it's still relevant. It is.
• [x] This is a single feature request (not multiple features)

Problem Statement

I run several Claude Code sessions at once in the desktop app's 2x2 split view. Every time the app starts I rebuild the grid by hand. The app saves the column widths and row split (desktop-frame.paneStore.v1), but not the panes themselves: with four panes on screen, extraPanesByMode.code is saved as []. On the next launch the grid is gone.

I also want more than one arrangement, for example a client-work grid and a home-lab grid, and a way to switch between them without closing and reopening sessions.

Proposed Solution

1. Restore the last layout on launch: grid shape, pane sizes, and which session was in each pane. Empty "new session" panes come back as empty panes.
2. Saved layouts: name the current arrangement and reopen it in one action from a menu or keyboard shortcut.
3. Workspace tabs (nice to have): a row of tabs in the Code view, each holding its own split-view grid. Switching tabs swaps the whole grid while every session keeps running.

Alternative Solutions

• Pop-out windows on separate monitors or virtual desktops. That works, but they're scattered and have to be rearranged each time.
• Running the claude CLI in Windows Terminal with saved pane layouts. That gives real tabs and layouts, but you lose the desktop app's interface.
• Sidebar groups help me find chats again, but the grid still has to be rebuilt.

Priority

Medium

Feature Category

Desktop app / UI

Use Case Example

1. Monday morning, launch the Claude desktop app.
2. The Code tab reopens with the same 2x2 grid as last time, each pane on the session it had.
3. Click the "Home lab" layout tab. A different 2x2 grid of sessions takes over the window, and the client-work sessions keep running behind it.

Additional Context

Windows 11 desktop app, Code tab, Opus 5.5 sessions. Related: #88020, #86925, #69993.