Add sidebar tabs and nonblocking side-chat creation (#59767)

## Why

Starting a side chat waits for fork and setup requests. Keep the workspace responsive during those requests and let users return to an existing side chat through the sidebar chooser.

## What changed

- Add a clickable `Side chat` tab and `+` chooser, plus the configurable `focus_sidebar_tabs` action bound to `leader t` by default.
- Create workspace side chats in the background, showing loading and failure states with `Enter` to retry and `Esc` to go back. Preserve opening prompts and avoid taking focus when the user has moved away.
- Retain inherited permission profiles and defer startup notifications until the side chat is registered on its owning backend.
- Quiet the parent composer while the chooser has focus and prevent completion popups or repeated activation keys from consuming chooser input.

## Testing

Extend tests and snapshots for loading and failure states, returning to an existing chat, draft preservation, inherited permissions, focus transitions, and deferred MCP startup notifications.

GitOrigin-RevId: 244bcd3f170819e2260f21dc1041cf8899fe048c

codex-rs/tui/src/keymap/bindings.rs

@@ -109,6 +109,7 @@ impl KeymapActionId {
                 self.action,
                 "focus_main_composer"
                     | "focus_sidebar_content"
+                    | "focus_sidebar_tabs"
                     | "hide_pane"
                     | "focus_pane_left"
                     | "focus_pane_right"
@@ -316,6 +317,7 @@ define_runtime_action_bindings! {
         focus_activity,
         focus_main_composer,
         focus_sidebar_content,
+        focus_sidebar_tabs,
         hide_pane,
         focus_pane_left,
         focus_pane_right,

codex-rs/tui/src/keymap/chords.rs

@@ -205,7 +205,7 @@ impl KeymapContextSet {
                 | "maximize_pane_width"
                 | "resize_panes" => OPEN_PANE,
                 "focus_main_composer" => MAIN_COMPOSER_FOCUS,
-                "focus_sidebar_content" => SIDE_PANE | SIDEBAR_CHOOSER,
+                "focus_sidebar_content" | "focus_sidebar_tabs" => SIDE_PANE | SIDEBAR_CHOOSER,
                 _ => SIDE_PANE,
             };
             return self.0 & required != 0;

codex-rs/tui/src/keymap/chords/leader_defaults.rs

@@ -47,6 +47,7 @@ pub(in crate::keymap) fn register(
         (Global, "resize_panes", Char('r'), "r"),
         (Global, "minimize_pane_width", Char(','), ","),
         (Global, "maximize_pane_width", Char('.'), "."),
+        (Global, "focus_sidebar_tabs", Char('t'), "t"),
     ] {
         if remaining == 0 {
             break;