When not monitored Tasks is reported is not something we are in control of that is done when the GC is collected. That will happen when there is a code reload when entering or exiting playmode.

A better way of doing task like that is to monitor the completion of the task either by making the callback async

public async void Start()
{
await AsyncTest();
}

or if you still want a fire and forget style setup a continuation that validates the success and failure of the task

public void Start()
{
AsyncTest()
.ContinueWith(FireForgetComplete);
}

private void FireForgetComplete(Task task)
{
if (task.IsFaulted)
{
Debug.LogException(obj.Exception?.InnerException);
}
}