feat(sidebar): improve running session state tracking

Add a running-session view to the sidebar, including header controls, running counts, empty states, and row-level processing indicators so active provider work is visible outside the current chat.

Hydrate running state after refresh through a status-only /api/providers/sessions/running endpoint backed by chatRunRegistry.listRunningRuns, then sync and poll the frontend processingSessions map from AppContent without attaching to chat streams or replaying messages.

Preserve fresh local processing entries during sync so newly sent messages are not cleared before the backend registry catches up, and clear completed sessions once the status endpoint no longer reports them.

Thread active session state through sidebar project/session components, show rotating loaders for processing sessions, and keep the running search mode expanded and filterable.

Fix optimistic local user-message dedupe so repeated prompts are only collapsed when a matching server echo appears from the same send window, preventing sent messages from disappearing until assistant completion.

Add registry test coverage for listing currently running app sessions.

Tests: npx eslint on changed files; npx tsc --noEmit -p tsconfig.json; npx tsc --noEmit -p server/tsconfig.json; npx tsx --tsconfig server/tsconfig.json --test server/modules/websocket/tests/chat-run-registry.test.ts.

src/components/sidebar/view/subcomponents/SidebarSessionItem.tsx

@@ -1,5 +1,5 @@
 import { useEffect, useRef } from 'react';
-import { Check, Edit2, Trash2, X } from 'lucide-react';
+import { Check, Edit2, Loader2, Trash2, X } from 'lucide-react';
 import type { TFunction } from 'i18next';
 
 import { Badge, Button, Tooltip } from '../../../../shared/view/ui';
@@ -13,6 +13,7 @@ type SidebarSessionItemProps = {
   project: Project;
   session: SessionWithProvider;
   selectedSession: ProjectSession | null;
+  isProcessing: boolean;
   currentTime: Date;
   editingSession: string | null;
   editingSessionName: string;
@@ -63,6 +64,7 @@ export default function SidebarSessionItem({
   project,
   session,
   selectedSession,
+  isProcessing,
   currentTime,
   editingSession,
   editingSessionName,
@@ -117,7 +119,7 @@ export default function SidebarSessionItem({
 
   return (
     <div className="group relative">
-      {sessionView.isActive && (
+      {!isProcessing && sessionView.isActive && (
         <div className="absolute left-0 top-1/2 -translate-x-1 -translate-y-1/2 transform">
           <Tooltip content={t('tooltips.activeSessionIndicator')} position="right">
             <div
@@ -134,7 +136,9 @@ export default function SidebarSessionItem({
           className={cn(
             'p-2 mx-3 my-0.5 rounded-md bg-card border active:scale-[0.98] transition-all duration-150 relative',
             isSelected ? 'bg-primary/5 border-primary/20' : '',
-            !isSelected && sessionView.isActive
+            !isSelected && isProcessing
+              ? 'border-border/60 bg-muted/20'
+              : !isSelected && sessionView.isActive
               ? 'border-green-500/30 bg-green-50/5 dark:bg-green-900/5'
               : 'border-border/30',
           )}
@@ -153,7 +157,13 @@ export default function SidebarSessionItem({
             <div className="min-w-0 flex-1">
               <div className="flex items-center gap-2">
                 <div className="truncate text-xs font-medium text-foreground">{sessionView.sessionName}</div>
-                {compactSessionAge && (
+                {isProcessing ? (
+                  <Tooltip content={t('tooltips.processingSessionIndicator', 'Processing session')} position="top">
+                    <span className="ml-auto flex h-5 w-5 flex-shrink-0 items-center justify-center rounded-md text-muted-foreground transition-opacity duration-200 group-hover:opacity-0">
+                      <Loader2 className="h-3 w-3 animate-spin" />
+                    </span>
+                  </Tooltip>
+                ) : compactSessionAge && (
                   <span className="ml-auto flex-shrink-0 text-[11px] text-muted-foreground">{compactSessionAge}</span>
                 )}
               </div>
@@ -187,15 +197,16 @@ export default function SidebarSessionItem({
           className={cn(
             'w-full justify-start p-2 h-auto font-normal text-left hover:bg-accent/50 transition-colors duration-200',
             isSelected && 'bg-accent text-accent-foreground',
+            !isSelected && isProcessing && 'bg-muted/20 hover:bg-accent/50',
           )}
           onClick={() => onSessionSelect(session, project.projectId)}
         >
           <div className="flex w-full min-w-0 items-start gap-2">
             <SessionProviderLogo provider={session.__provider} className="mt-0.5 h-3 w-3 flex-shrink-0" />
             <div className="min-w-0 flex-1">
-              <div className="flex items-center gap-2">
+              <div className={cn('flex items-center gap-2', isProcessing && 'pr-7')}>
                 <div className="truncate text-xs font-medium text-foreground">{sessionView.sessionName}</div>
-                {compactSessionAge && (
+                {!isProcessing && compactSessionAge && (
                   <span
                     className={cn(
                       'ml-auto flex-shrink-0 text-[11px] text-muted-foreground transition-opacity duration-200',
@@ -213,6 +224,19 @@ export default function SidebarSessionItem({
           </div>
         </Button>
 
+        {isProcessing && (
+          <div
+            role="status"
+            aria-label={t('tooltips.processingSessionIndicator', 'Processing session')}
+            className={cn(
+              'pointer-events-none absolute right-2 top-1/2 flex h-6 w-6 -translate-y-1/2 items-center justify-center rounded-md text-muted-foreground transition-opacity duration-200',
+              isEditing ? 'opacity-0' : 'group-hover:opacity-0',
+            )}
+          >
+            <Loader2 className="h-3.5 w-3.5 animate-spin" />
+          </div>
+        )}
+
         <div
           ref={editingContainerRef}
           className={cn(