fix(split-pane): style and dismiss the split-picker menu

.split-picker-menu/-item/-empty (created in openSplitPicker()) had zero CSS
and could not be dismissed except by picking an item — a default-path defect
since the Split button ships enabled to anyone who flips showSplitButton on.
Add CSS matching the sibling .run-mode-menu popover's look (floating-bg
backdrop blur, border, shadow, z-index 1000 above the header's 100), and
dismiss on outside click or Escape via the same one-shot listener pattern
session-ui.js already uses for its other transient popovers
(toggleCaseSettings(), toggleRunModeMenu()). Picking an item now routes
through the same _dismissSplitPicker() method as the outside-click/Escape
handlers, so the listeners never outlive the menu.

Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>

src/web/public/styles.css

@@ -18635,3 +18635,43 @@ html[data-session-list="sidebar"][data-sidebar="collapsed"] .btn-sidebar-toggle
   flex: 1 1 auto;
   min-height: 0;
 }
+
+/* Split-picker menu: a small popover listing sessions to split with, appended
+   to document.body and positioned `fixed` by openSplitPicker() (JS sets
+   top/right against the Split header button's own rect). z-index above the
+   header (100) with headroom to spare, matching the sibling .run-mode-menu
+   dropdown's 1000. Dismissed by openSplitPane()'s own inline onclick, or by
+   the click-outside/Escape listeners installed alongside it. */
+.split-picker-menu {
+  position: fixed;
+  z-index: 1000;
+  min-width: 200px;
+  max-width: 320px;
+  padding: 4px;
+  background: var(--floating-bg);
+  backdrop-filter: blur(20px);
+  -webkit-backdrop-filter: blur(20px);
+  border: 1px solid var(--control-border);
+  border-radius: 10px;
+  box-shadow: 0 8px 32px rgba(0, 0, 0, 0.5), 0 2px 8px rgba(0, 0, 0, 0.3);
+}
+
+.split-picker-item {
+  padding: 8px 10px;
+  border-radius: 6px;
+  cursor: pointer;
+  font-size: 13px;
+  white-space: nowrap;
+  overflow: hidden;
+  text-overflow: ellipsis;
+}
+
+.split-picker-item:hover {
+  background: var(--control-bg-hover, rgba(255, 255, 255, 0.08));
+}
+
+.split-picker-empty {
+  padding: 8px 10px;
+  font-size: 13px;
+  color: var(--text-muted);
+}

src/web/public/terminal-split.js

@@ -146,7 +146,7 @@ Object.assign(CodemanApp.prototype, {
       menu.innerHTML = candidates
         .map(
           (c) =>
-            `<div class="split-picker-item" data-session-id="${escapeHtml(c.id)}" onclick="app.openSplitPane(${escapeHtml(JSON.stringify(c.id))}); document.getElementById('splitPickerMenu')?.remove();">${escapeHtml(c.label)}</div>`
+            `<div class="split-picker-item" data-session-id="${escapeHtml(c.id)}" onclick="app.openSplitPane(${escapeHtml(JSON.stringify(c.id))}); app._dismissSplitPicker();">${escapeHtml(c.label)}</div>`
         )
         .join('');
     }
@@ -158,6 +158,31 @@ Object.assign(CodemanApp.prototype, {
       menu.style.top = `${rect.bottom + 4}px`;
       menu.style.right = `${window.innerWidth - rect.right}px`;
     }
+
+    // Dismiss on outside click or Escape — same one-shot listener pattern as
+    // session-ui.js's other transient popovers (toggleCaseSettings(),
+    // toggleRunModeMenu()). Deferred by a tick so the click that OPENED the
+    // menu (still bubbling) doesn't immediately close it. Picking an item
+    // (above) calls the SAME dismiss method, so these listeners never
+    // outlive the menu either way.
+    const closeOnOutsideClick = (e) => {
+      if (!menu.contains(e.target) && e.target !== splitBtn) this._dismissSplitPicker();
+    };
+    const closeOnEscape = (e) => {
+      if (e.key === 'Escape') this._dismissSplitPicker();
+    };
+    this._splitPickerDismissHandlers = { closeOnOutsideClick, closeOnEscape };
+    setTimeout(() => document.addEventListener('click', closeOnOutsideClick), 0);
+    document.addEventListener('keydown', closeOnEscape);
+  },
+
+  _dismissSplitPicker() {
+    document.getElementById('splitPickerMenu')?.remove();
+    if (this._splitPickerDismissHandlers) {
+      document.removeEventListener('click', this._splitPickerDismissHandlers.closeOnOutsideClick);
+      document.removeEventListener('keydown', this._splitPickerDismissHandlers.closeOnEscape);
+      this._splitPickerDismissHandlers = null;
+    }
   },
 
   openSplitPane(sessionId) {