Fix subagent MCP startup status settling (#37344)

## Why

Subagents can leave cached MCP servers deferred indefinitely, causing the TUI to keep showing MCP startup as running after every server that reported startup has settled.

## What changed

- Clear configured MCP startup expectations for active subagent threads so their status is driven by actual startup notifications.
- Preserve configured-server startup tracking for primary threads and side conversations.

## Testing

Add coverage for active and resumed subagents with deferred servers, plus the existing side-conversation behavior.

GitOrigin-RevId: 4d929a162dd067b4a76351652a764a79ce60f31f

codex-rs/tui/src/app/app_server_events.rs

@@ -18,6 +18,22 @@ use codex_app_server_protocol::ServerRequest;
 
 impl App {
     pub(super) fn refresh_mcp_startup_expected_servers_from_config(&mut self) {
+        if self
+            .current_displayed_thread_id()
+            .zip(self.primary_thread_id)
+            .is_some_and(|(thread_id, primary_thread_id)| {
+                self.agent_navigation.is_parent_owned(thread_id)
+                    || (thread_id != primary_thread_id
+                        && !self.side_threads.contains_key(&thread_id))
+            })
+        {
+            // Subagents can defer cached servers indefinitely, so only servers
+            // that actually report startup should keep their status running.
+            self.chat_widget
+                .set_mcp_startup_expected_servers(std::iter::empty());
+            return;
+        }
+
         let enabled_config_mcp_servers: Vec<String> = self
             .config
             .mcp_servers