Show live background task counts

server/src/session/mod.rs

@@ -272,6 +272,10 @@ pub struct SessionInfo {
     pub status: SessionStatus,
     pub last_activity: f64,
     pub created: f64,
+    /// The provider's latest measured number of live background tasks.
+    /// Absent until it reports one; absence is not a measured zero.
+    #[serde(skip_serializing_if = "Option::is_none")]
+    pub background_tasks: Option<usize>,
     /// How many subagents this session has started, from a directory
     /// listing rather than reading each one's status -- see
     /// `GET /sessions/{id}/subagents` for that. 0 when it has none, not
@@ -608,6 +612,7 @@ impl LiveSession {
             status: *self.shared.status.lock().unwrap(),
             last_activity: *self.shared.last_activity.lock().unwrap(),
             created: self.meta.created,
+            background_tasks: self.driver().and_then(|driver| driver.background_tasks()),
             subagents: subagent::count(self.dir()),
         }
     }
@@ -1140,6 +1145,7 @@ impl SessionManager {
                     status: status_of_unlaunched(&self.data_dir.join(&meta.id)),
                     last_activity: meta.created,
                     created: meta.created,
+                    background_tasks: None,
                     subagents: subagent::count(&self.data_dir.join(&meta.id)),
                 },
             })