Studio displays incorrect error when logging into a deleted Studio application

### Before opening, please confirm:

- [X] I have [searched for duplicate or closed issues](https://github.com/aws-amplify/amplify-adminui/issues?q=is%3Aissue+).
- [X] I have read the guide for [submitting bug reports](https://github.com/aws-amplify/amplify-adminui/blob/main/CONTRIBUTING.md#bug-reports).
- [X] I have done my best to include a minimal, self-contained set of instructions for consistently reproducing the issue.
- [X] I have removed any sensitive information from my code snippets and submission.

### App Id

d2d0dufan7sxoi

### Region

ap-southeast-2

### Environment name

web

### Figma File Version (if applicable)

_No response_

### Amplify CLI Version

_No response_

### If applicable, what version of Node.js are you using?

_No response_

### What operating system are you using?

_No response_

### Browser type?

_No response_

### Describe the bug

When I tried to:

- Turn on amplify studio
- Enter amplify studio
- Delete project

The error of 'network failure' occured

### Expected behavior

Should enter the amplify studio straightly

### Reproduction steps

1. Enter amplify console
2. Enter project
3. Enter amplify studio/delete project
4. Error occur

### Project Identifier

_No response_

### Additional information

_No response_

Start by reproducing the reported sequence in the Amplify console and Studio: enable Studio, enter the project, delete it, and observe the login response. Trace how Studio handles a deleted application and identify why it surfaces a network failure. Done means the deleted-application case produces the expected behavior instead of the reported error.

Written by the indexing model from the issue text.