Not sure if it's related, but my wife's GD didn't auto stable either. I saw her char logging fine but the GD remained.
She was fighting invasion monsters before that. Enough time has passed since her last kill that the combat timer didn't prevent her from insta-logout. But even not, the GD would have auto stabled when she logged.
Tried it a couple of times, in the end, just stabled the GD at the stable master.