[Bug]: All previously saved Connection information has been deleted.

### Description

Last night I shut down my computer. This morning I turned it on, and all my Connection information in the SQL Server extension in VS Code was gone.

I didn't install anything new, update anything, or delete anything.

My Query History records are still there. However, my Connection records are missing. The "SQL Server" window says "Add Connection," as if I were installing it for the first time.

How can I recover my existing Connection records?

### Steps to Reproduce

I've tried closing and reopening VS Code, but the result remains the same. I don't have the database Connection information.

### Affected Area

- [ ] 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
- [ ] SQL Database projects
- [ ] Query Plan Visualizer
- [x] Other (please describe below)

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

I'm having trouble with the "Connection" field in the SQL Server extension (add-in) window in VS Code.

### Environment Information

Version: 1.137.0 (Universal)
Commit: 645f29cc3176500b4b5762ba887cf2a7f0ffdf2c
Date: 2026-09-08T13:43:59-07:00 (7 days ago)
Electron: 42.10.0
ElectronBuildId: 15109253
Chromium: 148.0.7778.280
Node.js: 24.18.1
V8: 14.8.178.38-electron.0
OS: Darwin x64 24.6.0

### Confirmation

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

The report identifies the SQL Server extension's Connection field in VS Code and notes that Query History remains while Connection records disappear. No source files or tests are named, so begin by locating the connection-record persistence entry point and compare it with query-history handling. Done means the loss can be reproduced and existing Connection records remain available after restarting VS Code.

Written by the indexing model from the issue text.