feat add processing conversation on desktop

app/lib/desktop/pages/conversations/desktop_conversations_page.dart

@@ -21,6 +21,7 @@ import 'widgets/desktop_conversation_card.dart';
 import 'widgets/desktop_daily_score_widget.dart';
 import 'widgets/desktop_empty_conversations.dart';
 import 'widgets/desktop_goals_widget.dart';
+import 'widgets/desktop_processing_conversation_widget.dart';
 import 'widgets/desktop_recording_widget.dart';
 import 'widgets/desktop_search_result_header.dart';
 import 'widgets/desktop_search_widget.dart';
@@ -311,9 +312,18 @@ class _DesktopConversationsPageState extends State<DesktopConversationsPage>
                                       opacity: _fadeAnimation,
                                       child: Container(
                                         padding: const EdgeInsets.fromLTRB(32, 24, 32, 24),
-                                        child: DesktopRecordingWidget(
-                                          hasConversations: true,
-                                          onStartRecording: _showExpandedRecordingView,
+                                        child: Column(
+                                          children: [
+                                            DesktopRecordingWidget(
+                                              hasConversations: true,
+                                              onStartRecording: _showExpandedRecordingView,
+                                            ),
+                                            if (convoProvider.processingConversations.isNotEmpty) ...[
+                                              const SizedBox(height: 16),
+                                              getDesktopProcessingConversationsWidget(
+                                                  convoProvider.processingConversations),
+                                            ],
+                                          ],
                                         ),
                                       ),
                                     ),

app/lib/desktop/pages/conversations/widgets/desktop_processing_conversation_widget.dart

@@ -0,0 +1,99 @@
+import 'package:flutter/material.dart';
+
+import 'package:intl/intl.dart';
+
+import 'package:omi/backend/schema/conversation.dart';
+import 'package:omi/utils/l10n_extensions.dart';
+import 'package:omi/utils/responsive/responsive_helper.dart';
+
+class DesktopProcessingConversationWidget extends StatelessWidget {
+  final ServerConversation conversation;
+
+  const DesktopProcessingConversationWidget({
+    super.key,
+    required this.conversation,
+  });
+
+  @override
+  Widget build(BuildContext context) {
+    return Container(
+      padding: const EdgeInsets.all(16),
+      decoration: BoxDecoration(
+        color: const Color(0xFF1F1F25),
+        borderRadius: BorderRadius.circular(20),
+        border: Border.all(
+          color: ResponsiveHelper.backgroundTertiary.withValues(alpha: 0.5),
+          width: 1,
+        ),
+      ),
+      child: Column(
+        mainAxisSize: MainAxisSize.min,
+        crossAxisAlignment: CrossAxisAlignment.start,
+        children: [
+          // Header row
+          Row(
+            children: [
+              // Icon circle
+              Container(
+                width: 28,
+                height: 28,
+                decoration: BoxDecoration(
+                  color: ResponsiveHelper.backgroundQuaternary,
+                  borderRadius: BorderRadius.circular(14),
+                ),
+              ),
+              const SizedBox(width: 12),
+              // Processing label
+              Container(
+                decoration: BoxDecoration(
+                  color: const Color(0xFF35343B),
+                  borderRadius: BorderRadius.circular(20),
+                ),
+                padding: const EdgeInsets.symmetric(horizontal: 14, vertical: 8),
+                child: Text(
+                  context.l10n.processing,
+                  style: const TextStyle(
+                    color: Colors.white,
+                    fontSize: 14,
+                    fontWeight: FontWeight.w500,
+                  ),
+                ),
+              ),
+              const Spacer(),
+              // Timestamp
+              Text(
+                DateFormat.jm(Localizations.localeOf(context).languageCode)
+                    .format(conversation.startedAt ?? conversation.createdAt),
+                style: const TextStyle(
+                  color: ResponsiveHelper.textTertiary,
+                  fontSize: 13,
+                  fontWeight: FontWeight.w500,
+                ),
+              ),
+            ],
+          ),
+          const SizedBox(height: 14),
+          // Title placeholder bar
+          Container(
+            width: double.maxFinite,
+            height: 12,
+            decoration: BoxDecoration(
+              color: ResponsiveHelper.backgroundQuaternary,
+              borderRadius: BorderRadius.circular(6),
+            ),
+          ),
+        ],
+      ),
+    );
+  }
+}
+
+/// Widget to display processing conversations in a list
+Widget getDesktopProcessingConversationsWidget(List<ServerConversation> conversations) {
+  if (conversations.isEmpty) {
+    return const SizedBox.shrink();
+  }
+
+  // Show only the first (most recent) processing conversation
+  return DesktopProcessingConversationWidget(conversation: conversations.first);
+}