dask: Dashboard Badge Appears Before Readiness, Leading to "Page Not Found"

[Alputer]

The Dask dashboard badge is sometimes visible before the dashboard is fully ready, causing it to fall back to a Page Not Found error.

Previously, we attempted to fix this issue by switching the logic from checking the pod status to using the readiness_probe (reference). However, this behavior has been observed again in the recently deployed QA instance.

Let's investigate the issue further to ensure the dashboard badge only appears when the service is actually ready.