Q App Stuck on Welcome Screen after Sign-In

### Checks

- [x] I have searched [github.com/aws/amazon-q-developer-cli/issues](https://github.com/aws/amazon-q-developer-cli/issues?q=) and there are no duplicates of my issue
- [x] I have run `q doctor` in the affected terminal session
- [x] I have run `q restart` and replicated the issue again

### Operating system

macOS 15.6.1 (24G90)

### Expected behaviour

Clicking on the Amazon Q icon in the menu bar displaying the Welcome Screen and buttons for Sign In or Use with Pro license. After successfully signing in with authentication from browser. The welcome screen disappears and allows for editing the Q settings and features in the GUI for Q.

### Actual behaviour

After successfully signing in with authentication from browser the welcome screen remains visible preventing editing of settings or options.

### Steps to reproduce

- Open Terminal
- Run `q`
- Sign in via website interface
- Confirm Q is activated and functioning in terminal
- Run `q settings`
- Asked to sign in again on Q app interface welcome screen

### Environment

```yaml

[q-details]
version = "1.16.0"
hash = "34986d95f660e2bce63189f1179de4a110507fe7"
date = "2025-09-16T21:59:28.134585Z (21h ago)"
variant = "full"

[system-info]
os = "macOS 15.6.1 (24G90)"
chip = "Apple M3 Max"
total-cores = 14
memory = "36.00 GB"

[environment]
cwd = "/Users/USER/Sites/dupuy"
cli-path = "/Users/USER/Sites/dupuy"
os = "Mac"
shell-path = "/bin/zsh"
shell-version = "5.9"
terminal = "macOS"
install-method = "unknown"

[env-vars]
PATH = "/Users/USER/.local/bin:/Users/USER/.local/bin:/usr/local/share/npm/bin:/opt/homebrew/bin:/opt/homebrew/sbin:/usr/local/bin:/System/Cryptexes/App/usr/bin:/usr/bin:/bin:/usr/sbin:/sbin:/var/run/com.apple.security.cryptexd/codex.system/bootstrap/usr/local/bin:/var/run/com.apple.security.cryptexd/codex.system/bootstrap/usr/bin:/var/run/com.apple.security.cryptexd/codex.system/bootstrap/usr/appleinternal/bin:/Library/Apple/usr/bin:/Users/USER/.local/bin:/Users/USER/.lmstudio/bin:/Users/USER/.orbstack/bin:/Users/USER/.lmstudio/bin"
QTERM_SESSION_ID = "e0399dc0b525418b98d7df8fdaa94988"
Q_SET_PARENT_CHECK = "1"
Q_TERM = "1.16.0"
SHELL = "/bin/zsh"
TERM = "xterm-256color"
__CFBundleIdentifier = "com.apple.Terminal"
```

Riproduci il problema su macOS usando `q`, quindi verifica il comportamento con `q doctor`, `q restart` e `q settings` dopo l’accesso tramite browser. Non viene indicato alcun file sorgente né alcun test; segui il flusso di accesso desktop e determina se la schermata di benvenuto scompare dopo l’autenticazione, quindi aggiungi la copertura per la sequenza segnalata.

Scritto dal modello di indicizzazione a partire dal testo della issue.