Infinite Loading File Picker

### Godot version

v4.4.1.stable.official.49a5bc7b6

### VS Code version

Version: 1.101.0 (user setup) Commit: dfaf44141ea9deb3b4096f7cd6d24e00c147a4b1 Date: 2025-06-11T15:00:50.123Z

### Godot Tools VS Code extension version

2.5.1

### System information

Windows 11

### Issue description

When opening VSCode while Godot is already running, the file picker dialog will infinitely load:

Opening VSCode before opening Godot will make the file picker work properly:

`launch.json`:
```json
{
"version": "0.2.0",
"configurations": [
{
"name": "Godot",
"type": "godot",
"request": "launch",
"project": "${workspaceFolder}",
"port": 6007,
"debugServer": 6008,
"address": "127.0.0.1",
}
]
}
```
`settings.json`:
```json
{
"godotTools.editorPath.godot4": "c:\\Users\\Vadim\\Desktop\\Godot_v4.4.1-stable_win64.exe",
"files.exclude": {
"**/*.fbx": true,
"**/*.import": true,
"**/*.res": true,
"**/*.uid": true
},
"editor.detectIndentation": true,
"godotTools.inlayHints.gdscript": false,
"godotTools.lsp.autoReconnect.enabled": true,
"godotTools.lsp.serverPort": 6005
}
```

### Steps to reproduce

1. Open Godot
2. Open VSCode
3. Press `Ctrl+P`, type in a filename. Infinite loading.