fix(runner): defer activity caret until settled

ui/src/components/task-chat/TaskChatLiveRunPill.test.tsx

@@ -105,4 +105,28 @@ describe("TaskChatLiveRunPill", () => {
     expect(container.textContent).toContain("Workingfor 2s");
     expect(container.querySelector(".shimmer-text")).toBeNull();
   });
+
+  it("only exposes the runner disclosure caret after the run settles", () => {
+    const render = (status: string) => act(() => {
+      root.render(
+        <TaskChatLiveRunPill
+          status={status}
+          startedAtMs={1_000}
+          finishedAtMs={status === "succeeded" ? 3_000 : null}
+          toolSummary={null}
+          runnerStyle
+          expanded={false}
+          onToggle={() => undefined}
+        />,
+      );
+    });
+
+    render("running");
+    expect(container.querySelector('[data-testid="task-chat-live-run-pill"]')?.tagName).toBe("DIV");
+    expect(container.querySelector('[data-testid="task-chat-live-run-pill"] svg')).toBeNull();
+
+    render("succeeded");
+    expect(container.querySelector('[data-testid="task-chat-live-run-pill"]')?.tagName).toBe("BUTTON");
+    expect(container.querySelector('[data-testid="task-chat-live-run-pill"] svg')).not.toBeNull();
+  });
 });

ui/src/components/task-chat/TaskChatLiveRunPill.tsx

@@ -82,13 +82,15 @@ export function TaskChatLiveRunPill({
       <>
         <span className="text-sm font-medium text-foreground/80">{verb}</span>
         {suffix ? <span className="text-sm text-muted-foreground">{suffix}</span> : null}
-        <ChevronRight
-          className={cn("ml-auto h-3.5 w-3.5 text-muted-foreground transition-transform", expanded && "rotate-90")}
-          aria-hidden
-        />
+        {!active ? (
+          <ChevronRight
+            className={cn("ml-auto h-3.5 w-3.5 text-muted-foreground transition-transform", expanded && "rotate-90")}
+            aria-hidden
+          />
+        ) : null}
       </>
     );
-    return onToggle ? (
+    return onToggle && !active ? (
       <button
         type="button"
         onClick={onToggle}