[Bug]: I can't connect to my database while I have an SQL script open

### Description

### Steps to Reproduce

1. Open a new SQL script
2. Open a database connection

### Affected Area

- [x] Connection dialog (SQL Server | Azure browse/Fabric browse)
- [ ] Query editor
- [ ] Query results panel
- [ ] 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
- [x] SQL Database projects
- [ ] Query Plan Visualizer
- [ ] Other (please describe below)

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

_No response_

### Environment Information

- MSSQL EXTENSION VERSION
- Windows 11
- Cursor AI IDE

### Confirmation

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

Start with the connection dialog flow while an SQL script is open, focusing on the SQL Database projects area identified in the report. Reproduce the failure on Windows 11 in Cursor AI IDE, then trace the interaction between the open script and database connection. Done means a database connection can be opened successfully with a new SQL script present.

Written by the indexing model from the issue text.