feat: persist task lane across submits and session reloads

Task lane questions/actions are now saved to a pending_task_lane JSONB
column on ai_sessions, restoring them on session switch or page reload.
Partial submit no longer force-clears the lane — the AI response
controls what stays. Also removes redundant "New Session" button from
the sidebar (dashboard already provides this).

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

backend/app/services/unified_chat_service.py

@@ -286,6 +286,15 @@ async def send_chat_message(
                 except Exception:
                     logger.exception("Failed to create fork within branch for session %s", session.id)
 
+            # Persist task lane state on session
+            if branch_questions_data or branch_actions_data:
+                session.pending_task_lane = {
+                    "questions": branch_questions_data or [],
+                    "actions": branch_actions_data or [],
+                }
+            else:
+                session.pending_task_lane = None
+
             suggested_flows = extract_suggested_flows(
                 await rag_search(query=message, account_id=account_id, db=db, limit=8)
             )
@@ -393,6 +402,15 @@ async def send_chat_message(
             logger.exception("Failed to create fork for session %s", session_id)
             # Fork failed but chat message still sent — don't break the response
 
+    # Persist task lane state on session
+    if questions_data or actions_data:
+        session.pending_task_lane = {
+            "questions": questions_data or [],
+            "actions": actions_data or [],
+        }
+    else:
+        session.pending_task_lane = None
+
     suggested_flows = extract_suggested_flows(rag_results)
 
     return display_content, suggested_flows, session, fork_metadata, actions_data, questions_data