격려문구 알림 팝업 자동 종료 안정성 개선

- ReminderPopupWindow의 자동 닫힘 경로를 점검하고 UI 타이머와 실제 종료 타이머를 분리함
- 남은 시간 계산을 절대 시각 기준으로 바꾸고 Task.Delay 기반 종료 fail-safe를 추가해 팝업이 남는 경우를 줄임
- 창 종료 시 CancellationToken과 타이머를 함께 정리해 후속 종료 처리도 안전하게 맞춤
- README와 DEVELOPMENT 문서에 변경 목적과 검증 결과를 로컬 시각 기준으로 기록함
- 검증: dotnet build src/AxCopilot/AxCopilot.csproj -c Release -v minimal -p:OutputPath=bin\\verify\\ -p:IntermediateOutputPath=obj\\verify\\ (경고 0, 오류 0)

docs/DEVELOPMENT.md

@@ -4965,3 +4965,5 @@ ow + toggle ?쒓컖 ?몄뼱濡??ㅼ떆 ?뺣젹?덈떎.
 - Document update: 2026-04-06 15:35 (KST) - Reordered the AX Agent internal-settings common tab in `ChatWindow.xaml` so `서비스와 모델` and `등록 모델 관리` now appear consecutively. This matches the actual model-management flow more closely.
 - Document update: 2026-04-06 15:35 (KST) - Moved the `운영 모드` chooser below the conversation management/storage area and restored section separators around the moved blocks so the common-tab layout reads in clearer grouped sections.
 - Document update: 2026-04-06 15:41 (KST) - Reworked the empty-state layout in `ChatWindow.xaml` so the icon/title/description block and the preset grid live inside one vertically centered stack. This fixes the previous behavior where only the preset cards looked centered while the descriptive header stayed visually high on tall windows.
+- Document update: 2026-04-06 15:48 (KST) - Hardened the unlock-reminder popup auto-close path in `ReminderPopupWindow.xaml.cs`. The window now tracks an absolute close deadline and uses a separate `Task.Delay + CancellationToken` fail-safe to close the popup even if the UI timer tick is delayed.
+- Document update: 2026-04-06 15:48 (KST) - The countdown bar still updates through `DispatcherTimer`, but the actual close action is now guaranteed by the background delay path. This reduces cases where encouragement popups remain visible after the configured display duration.