feat(cursor): sync local conversation history to Cursor transcripts

Project sidecar context history into Cursor-compatible JSONL transcripts so previous conversations remain readable when referenced from new chats. Backfill existing sessions on startup and preserve Cursor-managed turn status entries during atomic updates.

internal/backend/forwarder/service.go

@@ -302,6 +302,7 @@ func NewService(historyRoot string, resolver modeladapter.ChannelResolver) *Serv
 		appendSeq:          newAppendSequenceTracker(),
 	}
 	service.startHistoryMaintenance()
+	store.SyncAllCursorTranscriptsBestEffort()
 	return service
 }
 
@@ -691,6 +692,11 @@ func (service *Service) handleRunIntent(intent InboundIntent) error {
 	if err != nil {
 		return err
 	}
+	if intent.RequestContext != nil {
+		if folder := normalizeAgentTranscriptsFolder(intent.RequestContext.GetEnv().GetAgentTranscriptsFolder()); folder != "" {
+			conversation.AgentTranscriptsFolder = folder
+		}
+	}
 	rewindDecision := service.decideRunRewind(intent, conversation)
 	if rewindDecision.Evaluated && !rewindDecision.Apply {
 		service.logRunRewindDecision(intent.RequestID, intent.ConversationID, "rewind_skipped", rewindDecision)