fix(conversation_store,admin,cli): keep session when file lags db, sort pty by recency

src/beaver_gateway/backends/claude_code.py

@@ -199,6 +199,7 @@ def _build_backend_options(
         startup_delay=opt.startup_delay,
         file_wait_timeout=opt.file_wait_timeout,
         turn_duration_timeout=opt.turn_duration_timeout,
+        idle_session_ttl=opt.idle_session_ttl,
     )