fix(webapp): show Idle session status on the run page badge too

The run/span panel derived the session badge from closedAt/expiresAt only, so it could read Active where the sessions list and detail page now read Idle. It now uses the same run-liveness derivation as the rest of the sessions surface.

Author: D-K-P

apps/webapp/app/presenters/v3/SpanPresenter.server.ts

@@ -36,6 +36,7 @@ import { runStore } from "~/v3/runStore.server";
 import { getTaskEventStoreTableForRun, type TaskEventStoreTable } from "~/v3/taskEventStore.server";
 import { isFailedRunStatus, isFinalRunStatus } from "~/v3/taskStatus";
 import { BasePresenter } from "./basePresenter.server";
+import { deriveSessionStatus } from "./deriveSessionStatus";
 import { WaitpointPresenter } from "./WaitpointPresenter.server";
 import {
   controlPlaneResolver,
@@ -358,25 +359,41 @@ export class SpanPresenter extends BasePresenter {
                 taskIdentifier: true,
                 closedAt: true,
                 expiresAt: true,
+                currentRunId: true,
               },
             },
           },
         })
       : null;
 
+    // Resolve the session's current run so the badge reflects run liveness
+    // (the same IDLE-vs-ACTIVE distinction as the sessions list/detail), not
+    // just closedAt/expiresAt. Env-scoped, matching the run reads here.
+    const sessionCurrentRun =
+      sessionRun && sessionRun.session.currentRunId
+        ? await runStore.findRun(
+            {
+              id: sessionRun.session.currentRunId,
+              runtimeEnvironmentId: run.runtimeEnvironmentId,
+            },
+            { select: { status: true } },
+            this._replica
+          )
+        : null;
+
     const session = sessionRun
       ? {
           friendlyId: sessionRun.session.friendlyId,
           externalId: sessionRun.session.externalId,
           type: sessionRun.session.type,
           taskIdentifier: sessionRun.session.taskIdentifier,
-          status:
-            sessionRun.session.closedAt != null
-              ? ("CLOSED" as const)
-              : sessionRun.session.expiresAt != null &&
-                  sessionRun.session.expiresAt.getTime() < Date.now()
-                ? ("EXPIRED" as const)
-                : ("ACTIVE" as const),
+          status: deriveSessionStatus({
+            closedAt: sessionRun.session.closedAt,
+            expiresAt: sessionRun.session.expiresAt,
+            hasCurrentRun: sessionRun.session.currentRunId != null,
+            currentRunStatus: sessionCurrentRun?.status,
+            now: Date.now(),
+          }),
           reason: sessionRun.reason,
           triggeredAt: sessionRun.triggeredAt,
         }