fix(app): say it out loud when wake-ups do not work

Registration failures from the distributor were swallowed, so push looked
enabled while nothing arrived. Without wake-ups Android suspends the app
and messages only show up once the user opens it — which looks like broken
delivery even though the relay has them queued.

The failure is now remembered and surfaced in Settings, and the chat list
carries a dismissible line explaining the state.

app/lib/features/chat_list/chat_list_screen.dart

@@ -5,6 +5,7 @@ import 'package:sepot/core/l10n/error_messages.dart';
 import 'package:sepot/core/l10n/l10n.dart';
 import 'package:sepot/core/theme/app_theme.dart';
 import 'package:sepot/features/chat_list/chat_list_tile.dart';
+import 'package:sepot/features/chat_list/push_hint_banner.dart';
 import 'package:sepot/models/conversation.dart';
 import 'package:sepot/providers/contacts_provider.dart';
 import 'package:sepot/providers/conversations_provider.dart';
@@ -62,7 +63,13 @@ class ChatListScreen extends ConsumerWidget {
           }
           final hasArchive = archived.isNotEmpty;
 
-          return ListView.builder(
+          return Column(
+            children: [
+              // Bez budíčků appka na pozadí spí — ať uživatel ví, proč
+              // mu zprávy chodí, až když Šepot otevře.
+              const PushHintBanner(),
+              Expanded(
+                child: ListView.builder(
             padding: const EdgeInsets.symmetric(vertical: 8),
             itemCount: active.length + (hasArchive ? 1 : 0),
             itemBuilder: (context, index) {
@@ -95,7 +102,10 @@ class ChatListScreen extends ConsumerWidget {
                 onTap: () => context.push('/conversation/${conv.contactId}'),
                 onLongPress: () => _showChatActions(context, ref, conv, name),
               );
-            },
+                  },
+                ),
+              ),
+            ],
           );
         },
         loading: () => const Center(

app/lib/features/chat_list/push_hint_banner.dart

@@ -0,0 +1,73 @@
+import 'package:flutter/material.dart';
+import 'package:flutter_riverpod/flutter_riverpod.dart';
+import 'package:go_router/go_router.dart';
+import 'package:sepot/core/l10n/l10n.dart';
+import 'package:sepot/core/theme/app_theme.dart';
+import 'package:sepot/providers/push_provider.dart';
+import 'package:sepot/providers/storage_provider.dart';
+
+/// Upozornění nad seznamem chatů, když budíčky nefungují.
+///
+/// Bez pushe Android appku na pozadí uspí a zprávy dorazí, až ji uživatel
+/// sám otevře — vypadá to jako výpadek doručování, i když relay má všechno
+/// připravené ve frontě. Proužek je proto informace o stavu, ne reklama:
+/// zmizí, jakmile si push nastaví, a jde odmítnout natrvalo.
+class PushHintBanner extends ConsumerWidget {
+  const PushHintBanner({super.key});
+
+  @override
+  Widget build(BuildContext context, WidgetRef ref) {
+    final health = ref.watch(pushHealthProvider).valueOrNull;
+    final dismissed = ref.watch(pushHintDismissedProvider).valueOrNull ?? true;
+    if (dismissed || health == null || health == PushHealth.ok) {
+      return const SizedBox.shrink();
+    }
+
+    final text = health == PushHealth.failed
+        ? context.l10n.pushHintFailed
+        : context.l10n.pushHintOff;
+
+    return Material(
+      color: SepotColors.surfaceVariant,
+      child: InkWell(
+        onTap: () => context.push('/settings'),
+        child: Padding(
+          padding: const EdgeInsets.fromLTRB(16, 10, 8, 10),
+          child: Row(
+            children: [
+              Icon(
+                Icons.notifications_off_outlined,
+                size: 18,
+                color: health == PushHealth.failed
+                    ? SepotColors.ember
+                    : SepotColors.textMuted,
+              ),
+              const SizedBox(width: 12),
+              Expanded(
+                child: Text(
+                  text,
+                  style: const TextStyle(
+                    fontFamily: kFontSans,
+                    fontSize: 13,
+                    color: SepotColors.textSecondary,
+                  ),
+                ),
+              ),
+              IconButton(
+                icon: const Icon(Icons.close, size: 18),
+                color: SepotColors.textMuted,
+                tooltip: context.l10n.commonClose,
+                onPressed: () async {
+                  await ref
+                      .read(storageProvider)
+                      .setSetting('push_hint_dismissed', true);
+                  ref.invalidate(pushHintDismissedProvider);
+                },
+              ),
+            ],
+          ),
+        ),
+      ),
+    );
+  }
+}