[BACKGROUND TASK READY] never goes away

### 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 15.7.1 (24G231)

### Expected behaviour

When running background (delegate) tasks, receive the answer and then discard what was done.

### Actual behaviour

I once ran a `delegate` task inside one of my Q sessions and it has never stopped telling me that `[BACKGROUND TASK READY]`. I have no way to remove it.

### Steps to reproduce

_No response_

### Environment

```yaml

[q-details]
version = "1.19.3"
hash = "9146a5c9dd3fafcfb4eda5f27f7e3a27a002ab34"
date = "2025-10-29T20:38:27.970829Z (2d ago)"

[system-info]
os = "macOS 15.7.1 (24G231)"
chip = "Apple M4 Pro"
total-cores = 14
memory = "48.00 GB"

[environment]
cwd = "/Users/USER/Documents/Code"
cli-path = "/Users/USER/Documents/Code"
install-method = "unknown"

[env-vars]
PATH = "/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/TeX/texbin:/opt/homebrew/bin:/Users/USER/.toolbox/bin:/Users/USER/.sdkman/candidates/java/current/bin:/opt/homebrew/opt/libpq/bin:/opt/homebrew/opt/llvm/bin:/Users/USER/.nvm/versions/node/v22.19.0/bin:/opt/homebrew/sbin:/Applications/iTerm.app/Contents/Resources/utilities:/Users/USER/.local/bin:/Users/USER/Library/Application Support/JetBrains/Toolbox/scripts:/Users/USER/.dotnet/tools"
QTERM_SESSION_ID = "5dfb40a01e5441a0bdd04e5b6bbeebf0"
Q_SET_PARENT_CHECK = "1"
Q_TERM = "1.19.3"
SHELL = "/bin/zsh"
TERM = "xterm-256color"
__CFBundleIdentifier = "com.googlecode.iterm2"
```

Empieza reproduciendo la notificación persistente `[BACKGROUND TASK READY]` con una tarea `delegate` en la sesión de terminal afectada, usando `q doctor` y `q restart` tal como se ha informado. Rastrea cómo se conserva y se muestra el resultado de la tarea en segundo plano; el trabajo estará terminado cuando se pueda recibir la respuesta y la tarea completada se descarte sin que la notificación vuelva a aparecer.

Escrito por el modelo de indexación a partir del texto del issue.