The prompt "Waiting for Azure sign-in…" doesn't disappear after completing the sign-in

OS: Win 10
Build Version: 20230531.1
Regression: Not a Regression

Repro Steps:

1. Open a dist folder in VS Code.
2. Create a storage account.
3. F1 -> Sign Out.
4. F1 -> Azure: Sign In with Device Code -> Click "Copy & Open" (Don't paste the code into the webpage yet).
5. Back to VS Code -> Navigate to "Explorer" view -> Right click the folder -> Click "Deploy to Static Website via Azure Storage…".
6. Pop up "Waiting for Azure sign-in…".
   
7. Back to the sign-in page in browser --> Enter the code and complete the sign-in.
8. Check whether the prompt "Waiting for Azure sign-in…" disappears.

Expect:
The prompt "Waiting for Azure sign-in…" disappears.

Actual:
The prompt "Waiting for Azure sign-in…" doesn't disappear.

More Info:

1. The subscriptions list in the RESOURCES view after executing the above steps.
2. Sign in again after clicking the prompt "Waiting for Azure sign-in…".

Start by reproducing the issue in VS Code through the "Deploy to Static Website via Azure Storage…" entry point while completing Azure device-code sign-in in the browser. Trace the extension's waiting prompt and sign-in flow; done means the "Waiting for Azure sign-in…" prompt disappears immediately after sign-in completes.