fix(workspace): keep the console cwd on the selected lane's own folder

The cwd was synced only when laneId changed, but a pane can render before
GET /api/lanes has answered — split view restores its pane lanes from
localStorage, and layout 1 paints before the list loads. With no lane
resolved the field fell back to the home suggestion, and since laneId never
changed afterwards it stayed there. RunSetup submits that string verbatim to
POST /api/lanes/:id/start, so the run was started in the wrong folder.

Track the cwd in its own effect keyed on the resolved lane path rather than
on laneId, so it re-syncs as soon as the lane is known. The home default now
applies only while no lane is selected at all.

client/src/pages/__tests__/Workspace.laneCwd.test.tsx

@@ -0,0 +1,156 @@
+/**
+ * @file Workspace.laneCwd.test.tsx
+ * @description The console's working directory must be the selected lane's own
+ * `cwd` — on first paint (the page auto-selects the first lane) and after every
+ * lane switch. A stale cwd is not cosmetic: `RunSetup` submits it verbatim, so
+ * the run would be started in the previously selected lane's folder.
+ * @author Nguyễn Ngọc Trí Vĩ <vinnt@smartgift.vn>
+ */
+
+import { describe, it, expect, vi, beforeEach, afterEach } from "vitest";
+import { render, act, screen, fireEvent } from "@testing-library/react";
+import { MemoryRouter } from "react-router-dom";
+import i18n from "i18next";
+
+// Hoisted: `vi.mock`'s factory is lifted above the module body, so the
+// fixtures it reads have to be lifted with it.
+const { HOME, LANE_A, LANE_B } = vi.hoisted(() => {
+  const laneStub = (id: number, cwd: string) => ({
+    id,
+    title: `lane-${id}`,
+    cwd,
+    branch: null,
+    kind: "adopted",
+    source_repo: null,
+    pipeline: "default",
+    session_id: null,
+    run_id: null,
+    stage: "idle",
+    stage_since: null,
+    status: "idle",
+    gate_decision: null,
+    ci_status: null,
+    needs_action: null,
+    links: {},
+    stages: {},
+    notes: null,
+    pipeline_name: "Default",
+    pipeline_nodes: [],
+    progress: 0,
+    stage_seconds: null,
+    last_event_seconds: null,
+    liveness: "idle",
+    detected_stage: null,
+    detected_signal: null,
+    slot: null,
+    ports: {},
+    active_feature_id: null,
+  });
+  return {
+    HOME: { kind: "home", path: "/Users/tester", label: "Home" },
+    LANE_A: laneStub(1, "/workspace/alpha"),
+    LANE_B: laneStub(2, "/workspace/beta"),
+  };
+});
+
+vi.mock("../../lib/api", async (importOriginal) => {
+  const actual = await importOriginal<Record<string, unknown>>();
+  const r = (value: unknown) => vi.fn().mockResolvedValue(value);
+  return {
+    ...actual,
+    api: {
+      run: {
+        list: r({ items: [] }),
+        history: r({ items: [] }),
+        binary: r({ found: true, path: "/usr/bin/claude" }),
+        cwds: r({ items: [HOME] }),
+        files: r({ items: [] }),
+        start: r({ id: "run-1", status: "running" }),
+        get: r({ id: "run-1", status: "running" }),
+      },
+      lanes: {
+        list: r({
+          lanes: [LANE_A, LANE_B],
+          counts: { total: 2, running: 0, needs_you: 0, dead: 0 },
+        }),
+        pipelines: r({ pipelines: [] }),
+        git: r({ available: false }),
+        runtime: r({ up: false, ports: {}, lastError: null }),
+        features: { list: r({ features: [] }), show: r({ feature: null }) },
+        proof: { list: r({ features: [] }), imageUrl: () => "" },
+      },
+      sessions: { list: r({ sessions: [], total: 0, limit: 50, offset: 0 }) },
+    },
+  };
+});
+
+vi.mock("../../lib/eventBus", () => ({
+  eventBus: {
+    subscribe: () => () => {},
+    publish: () => {},
+    onConnection: () => () => {},
+    connected: true,
+    setConnected: () => {},
+  },
+}));
+
+import { Workspace } from "../Workspace";
+
+class ObserverStub {
+  observe() {}
+  unobserve() {}
+  disconnect() {}
+  takeRecords() {
+    return [];
+  }
+}
+globalThis.ResizeObserver =
+  globalThis.ResizeObserver || (ObserverStub as unknown as typeof ResizeObserver);
+
+async function settle() {
+  await act(async () => {
+    await new Promise((r) => setTimeout(r, 0));
+    await new Promise((r) => setTimeout(r, 0));
+  });
+}
+
+function cwdInput(): HTMLInputElement {
+  return screen.getByPlaceholderText(i18n.t("run:fields.cwdPlaceholder")) as HTMLInputElement;
+}
+
+beforeEach(() => {
+  i18n.changeLanguage("en");
+});
+
+afterEach(() => {
+  vi.clearAllMocks();
+});
+
+describe("Workspace — the console cwd follows the selected lane", () => {
+  it("shows the auto-selected first lane's cwd on load, not the home default", async () => {
+    render(
+      <MemoryRouter initialEntries={["/run"]}>
+        <Workspace />
+      </MemoryRouter>
+    );
+    await settle();
+    expect(cwdInput().value).toBe(LANE_A.cwd);
+  });
+
+  it("swaps the cwd when another lane is selected in the strip", async () => {
+    render(
+      <MemoryRouter initialEntries={["/run"]}>
+        <Workspace />
+      </MemoryRouter>
+    );
+    await settle();
+
+    fireEvent.click(screen.getByTestId("lane-tile-2"));
+    await settle();
+    expect(cwdInput().value).toBe(LANE_B.cwd);
+
+    fireEvent.click(screen.getByTestId("lane-tile-1"));
+    await settle();
+    expect(cwdInput().value).toBe(LANE_A.cwd);
+  });
+});