Preserve running session replay state

frontend/app/components/assistant-ui/thread-list.tsx

@@ -50,6 +50,7 @@ export type ClawRunStatus = {
 	session_id?: string;
 	status?: "idle" | "queued" | "running" | "completed" | "failed" | "cancelled";
 	current_stage?: string;
+	started_at?: number;
 };
 
 type ClawStoredToolCall = {
@@ -431,7 +432,11 @@ export function toReplayRepository(
 			id,
 			role: "assistant",
 			parts,
-			metadata: { sessionId: session.session_id ?? fallbackSessionId },
+			metadata: {
+				sessionId: session.session_id ?? fallbackSessionId,
+				runId: runStatus.run_id,
+				runStartedAtMs: normalizeRunTimestampMs(runStatus.started_at),
+			},
 		} as UIMessage;
 		const threadMessage = {
 			id,
@@ -444,7 +449,11 @@ export function toReplayRepository(
 				unstable_annotations: [],
 				unstable_data: [],
 				steps: [],
-				custom: { sessionId: session.session_id ?? fallbackSessionId },
+				custom: {
+					sessionId: session.session_id ?? fallbackSessionId,
+					runId: runStatus.run_id,
+					runStartedAtMs: normalizeRunTimestampMs(runStatus.started_at),
+				},
 			},
 		} as ThreadMessage;
 		bindExternalStoreMessage(threadMessage, uiMessage);
@@ -463,6 +472,11 @@ function isActiveRunStatus(
 	return runStatus?.status === "queued" || runStatus?.status === "running";
 }
 
+function normalizeRunTimestampMs(value: unknown) {
+	if (typeof value !== "number" || !Number.isFinite(value)) return undefined;
+	return value < 10_000_000_000 ? Math.round(value * 1000) : Math.round(value);
+}
+
 function findLastAssistantContentIndex(messages: readonly ClawStoredMessage[]) {
 	for (let index = messages.length - 1; index >= 0; index -= 1) {
 		const message = messages[index];