app-server: keep the model cache warm (#28699)

## Why

The app server is long-lived, but its shared model cache otherwise
refreshes only when a caller needs it. Once the five-minute cache
expires, starting a thread or calling `model/list` can wait for
`/models` on the request path.

Refresh the cache in the background before it expires so foreground
callers normally use fresh local state.

## What changed

- Start an app-server worker that refreshes models immediately and then
every three minutes using the existing models-manager API.
- Hold only a weak reference to the models manager between refreshes, so
the worker does not extend its lifetime.
- Stop scheduling refreshes when the app-server lifecycle handle is shut
down or dropped. A refresh already in progress is allowed to finish.
- Adjust affected app-server test fixtures to distinguish the background
`/models` probe from the connection they are testing.

The existing models-manager cache, refresh strategies, auth handling,
ETag behavior, and concurrency semantics are unchanged.

## Testing

-
`models_refresh_worker::tests::refreshes_immediately_periodically_and_stops_when_dropped`
-
`suite::v2::remote_control::listen_off_honors_persisted_remote_control_enable`
-
`suite::v2::attestation::attestation_generate_round_trip_adds_header_to_responses_websocket_handshake`

codex-rs/app-server/src/models_refresh_worker.rs

@@ -0,0 +1,65 @@
+use std::sync::Arc;
+use std::time::Duration;
+
+use codex_models_manager::manager::RefreshStrategy;
+use codex_models_manager::manager::SharedModelsManager;
+use tokio::task::JoinHandle;
+use tokio_util::sync::CancellationToken;
+
+const MODELS_REFRESH_INTERVAL: Duration = Duration::from_secs(3 * 60);
+
+#[derive(Debug)]
+pub(crate) struct ModelsRefreshWorker {
+    shutdown: CancellationToken,
+    _task: JoinHandle<()>,
+}
+
+impl ModelsRefreshWorker {
+    pub(crate) fn shutdown(&self) {
+        self.shutdown.cancel();
+    }
+}
+
+impl Drop for ModelsRefreshWorker {
+    fn drop(&mut self) {
+        self.shutdown();
+    }
+}
+
+pub(crate) fn spawn(models_manager: &SharedModelsManager) -> ModelsRefreshWorker {
+    spawn_with_interval(models_manager, MODELS_REFRESH_INTERVAL)
+}
+
+fn spawn_with_interval(
+    models_manager: &SharedModelsManager,
+    refresh_interval: Duration,
+) -> ModelsRefreshWorker {
+    let models_manager = Arc::downgrade(models_manager);
+    let shutdown = CancellationToken::new();
+    let worker_shutdown = shutdown.clone();
+    let task = tokio::spawn(async move {
+        loop {
+            if worker_shutdown.is_cancelled() {
+                break;
+            }
+            let Some(models_manager) = models_manager.upgrade() else {
+                break;
+            };
+            models_manager.list_models(RefreshStrategy::Online).await;
+            drop(models_manager);
+
+            tokio::select! {
+                _ = worker_shutdown.cancelled() => break,
+                _ = tokio::time::sleep(refresh_interval) => {}
+            }
+        }
+    });
+    ModelsRefreshWorker {
+        shutdown,
+        _task: task,
+    }
+}
+
+#[cfg(test)]
+#[path = "models_refresh_worker_tests.rs"]
+mod tests;