[Bug]: Unable to sync settings VS Code

### Description

Every time that I open VSCode in Windows and go to the mssql extension, it takes minutes to load before it gives me a message saying that my settings file is not valid. I have tried reinstalling, and I don't have this issue w/ my Linux installations of VSCode-mssql. I suspect it's a combination of extensions.

### Steps to Reproduce

- Open VSCode (windows)
- Select SQL Server in extension list (or Ctrl+Alt+D)
- Watch spinners for 2-3 minutes
- Look for notification saying: Unable to sync settings because the content in the file is not valid. Please open the file and correct.
- Open file and see that it is in fact corrupt:

```json
{
"mssql.connectionGroups": "[
{
"name": "ROOT"",
"id": ""ROOT"
}
]",
"git.confirmSync": false,
"files.autoSave": "afterDelay",
"http.proxy": "",
"http.proxyStrictSSL": true
}
```

- Fix the file
- Create connections
- Repeat...

### Affected Area

- [ ] Connection dialog (SQL Server | Azure browse/Fabric browse)
- [ ] Query editor
- [ ] Query results panel
- [x] Object Explorer
- [ ] GitHub Copilot integration
- [ ] Preview/Edit data
- [ ] Table Designer
- [ ] Schema Designer
- [ ] Schema Compare
- [ ] Local SQL Server Container provisioning
- [ ] SQL database in Fabric provisioning
- [ ] DACPAC/BACPAC export/import
- [ ] SQL Database projects
- [ ] Query Plan Visualizer
- [ ] Other (please describe below)

### If you selected "Other", please describe the affected area

_No response_

### Environment Information

- VS Code Version: 1.137.0 (user)
- MSSQL Extension Version: 1.45.1
- OS: Win 11 Ent 25H2 build 26200.9445
- Connection Type: (N/A)

### Confirmation

- [x] I have searched existing issues and couldn't find a match
- [ ] I want to work on this issue

Start by reproducing the issue on Windows with VS Code 1.137.0 and MSSQL extension 1.45.1, then inspect the generated settings file and compare the behavior with Linux. The fix is complete when opening Object Explorer no longer corrupts the settings JSON or reports that settings synchronization is invalid.

Written by the indexing model from the issue text.