Confiture,
well at 20000' it may be recoverable. But maybe is a pretty bad word in the way I operate an airliner.
At FL350, as the boys wrote, it might dive you into a highspeed stall, intervene with traffic below. You have people walking, trolleys loose, hot drinks being served, and you would risk the jerk of a sudden intercept? Quite dangerous, I'd say.
What's so hard to understand about this problem? The ILS reciever becomes "hot" upon pressing that button, AND having an ILS approach programmed into the FMS. Any signal of the same frequency as the preprogrammed ILS will trigger an intercept, if strong enough. The FMS and the ILS reciever are intermingled on this aircraft.
No toy, I'll stick with that!
Nic