[Bug] Intent Caching and BottomSheet Stacking via Android Widgets & Deep Links

### Describe your issue

When the application is launched from the background via an Android Home Screen Widget or Quick Action intent, GetX state management encounters two severe routing and context synchronization bugs:

1. **The Widget Controller Cache Loop:** Tapping "Task 1" on a widget correctly loads the Detail Route and instantiates a `DetailRouteController` for Task 1. However, if the app is backgrounded and the user taps a widget for "Task 3", GetX recognizes that `DETAIL_ROUTE` is already active in memory. Instead of routing the new UUID, GetX lazily reuses the cached `DetailRouteController`, keeping "Task 1" permanently stuck on screen for all subsequent widget taps.

2. **The "Black Screen" Context Void:** The Floating Action Button natively launched `showModalBottomSheet`. If a user tapped a "Quick Action" or "Widget Add Task" intent while the app was backgrounded with a native sheet open, GetX was unable to detect the native sheet via `Get.isBottomSheetOpen`. GetX stacked the new intent sheet on top. Upon saving, the system popped the GetX sheet, stranding the user inside an orphaned/black void natively.

### Steps to reproduce

**Widget Caching Loop:**
1. Tap Task 1 from an Android Widget.
2. Press the Android Home button to background the app.
3. Tap Task 3 from an Android Widget.
**Result:** The app incorrectly displays Task 1.

**Black Screen Void:**
1. Tap the in-app `+` FAB to open the Add Task modal.
2. Press the Android Home button to background the app.
3. Long-press the App Icon and trigger the "Add Task" Quick Action intent.
4. Fill out the task details and hit Save.
**Result:** The UI dismisses the active modal but reveals a broken/black overlay instead of the Home screen.

### What was the expected result?

The app should cleanly unmount stale overlays (like the native FAB bottom sheet) and explicitly purge legacy controllers (`DetailRouteController`) from memory before injecting new routing intents from a warm start. The new task should load accurately, and closing a bottom sheet should safely return the user to the underlying UI without crashing into a black screen.

### Put here any screenshots or videos (optional)

https://github.com/user-attachments/assets/5d021ebe-5f77-475e-b0d0-e6d84e01a383

### How can we contact you (optional)

_No response_

### Would you like to work on this issue?

Yes

### By submitting this issue, I have confirmed that:

- [x] I have starred the repo ⭐ and watched 👀 it on GitHub and followed the contribution guidelines.

まず、Android Home Screen Widget と Quick Action/deep-link の Flutter へのエントリーポイントをたどり、次に DetailRouteController と FAB のネイティブな showModalBottomSheet フローを調べます。バックグラウンドからの起動シナリオを両方再現し、新しい task が正しく開くこと、および intent 駆動の sheet を保存すると黒いオーバーレイなしで基盤となる Home 画面に戻ることを確認します。

索引モデルが issue の本文から書いたものです。