TUI support for buffer experience (#29919)

codex-rs/tui/src/chatwidget/replay.rs

@@ -24,6 +24,7 @@ impl ChatWidget {
                 duration_ms,
             } = turn;
             if matches!(status, TurnStatus::InProgress) {
+                self.turn_lifecycle.last_turn_id = Some(turn_id.clone());
                 self.last_non_retry_error = None;
                 self.on_task_started();
             }