feat(FN-1149): add send-to-background actions for AI planning modals

- Add Send to background header action to Planning Mode, Subtask Breakdown, and Mission Interview modals using a shared minimize icon pattern
- Show the action only during active AI session states and keep it hidden in initial states
- Implement background behavior to close the live stream connection and modal without canceling the underlying AI session
- Add modal tests that verify button visibility rules and confirm send-to-background does not trigger cancel APIs
- Add shared CSS styling for modal send-to-background controls, including hover and focus-visible states

packages/dashboard/app/components/SubtaskBreakdownModal.test.tsx

@@ -70,6 +70,19 @@ describe("SubtaskBreakdownModal", () => {
     expect(await screen.findByText("AI is generating subtasks...")).toBeInTheDocument();
   });
 
+  it("hides send to background button in initial state", () => {
+    render(
+      <SubtaskBreakdownModal
+        isOpen={true}
+        onClose={onClose}
+        initialDescription=""
+        onTasksCreated={onTasksCreated}
+      />,
+    );
+
+    expect(screen.queryByLabelText("Send to background")).not.toBeInTheDocument();
+  });
+
   it("renders editable subtasks when stream returns items", async () => {
     renderModal();
     await waitFor(() => expect(streamHandlers).toBeDefined());
@@ -181,6 +194,24 @@ describe("SubtaskBreakdownModal", () => {
     expect(onClose).toHaveBeenCalled();
   });
 
+  it("sends active breakdown session to background without canceling", async () => {
+    const closeSpy = vi.fn();
+    mockConnectSubtaskStream.mockImplementationOnce((_sessionId, _projectId, handlers) => {
+      streamHandlers = handlers;
+      return { close: closeSpy, isConnected: () => true };
+    });
+
+    renderModal();
+    await waitFor(() => expect(mockStartSubtaskBreakdown).toHaveBeenCalled());
+    await screen.findByText("AI is generating subtasks...");
+
+    fireEvent.click(screen.getByLabelText("Send to background"));
+
+    expect(closeSpy).toHaveBeenCalledTimes(1);
+    expect(mockCancelSubtaskBreakdown).not.toHaveBeenCalled();
+    expect(onClose).toHaveBeenCalledTimes(1);
+  });
+
   it("cancel closes modal", async () => {
     renderModal();
     await waitFor(() => expect(mockStartSubtaskBreakdown).toHaveBeenCalled());