fix(tui): label only detached subagents as background (#37306)

packages/tui/test/cli/tui/inline-tool-wrap-snapshot.test.tsx

@@ -4,6 +4,7 @@ import { testRender, type JSX } from "@opentui/solid"
 import {
   formatSubagentRetry,
   InlineToolRow,
+  isBackgroundSubagent,
   parseApplyPatchFiles,
   parseDiagnostics,
   parseQuestionAnswers,
@@ -201,6 +202,13 @@ describe("TUI inline tool wrapping", () => {
     expect(formatSubagentRetry(2, "Rate limited by provider")).toBe("Retrying (attempt 2) · Rate limited by provider")
   })
 
+  test("labels only detached or async subagents as background", () => {
+    expect(isBackgroundSubagent({ status: "running" }, "running")).toBeFalse()
+    expect(isBackgroundSubagent({ status: "running" }, "completed")).toBeTrue()
+    expect(isBackgroundSubagent({ status: "running" }, "error")).toBeFalse()
+    expect(isBackgroundSubagent({ status: "completed" }, "completed")).toBeFalse()
+  })
+
   test("snapshots consecutive grep, glob, and read rows at a narrow width", async () => {
     expect(await renderFrame(() => <Fixture />, { width: 72, height: 12 })).toMatchSnapshot()
   })