Dashboard error badge count disappears after switching tabs

## Describe the bug

When the dashboard shows an error badge with a non-zero count, switching between dashboard tabs causes the badge count to disappear even though the underlying error has not been resolved.

## Expected Behavior

The error badge count should persist across tab navigation and remain visible until the underlying errors are cleared.

## Steps To Reproduce

1. Open the Aspire dashboard in a state where an error badge count is displayed.
2. Switch to another dashboard tab.
3. Navigate back or continue switching tabs.
4. Observe that the error badge count has disappeared.

## Anything else?

The behavior was captured in a screen recording and screenshots in the originating Teams conversation.

_Created from a [Microsoft Teams conversation](https://teams.microsoft.com/l/message/19%3AuuDJgMCR7JeFSM-0RK2xRW5mAq2WFNfUK0TLmYDvAU01%40thread.tacv2/1788275061082?tenantId=72f988bf-86f1-41af-91ab-2d7cd011db47&groupId=f2e0b0b6-f130-46fa-8ec6-2ef4a7168ed2&parentMessageId=1788275061082)._