Yes, raise some weight in that particular case. The fact your head rises when horizontal indicates you're head-light / foot-heavy.I totally get that but if I go horizontal and motionless and my head rises you say "shift weight toward the end that rises" so I should move weight toward my head. Really!?
I suspect your "Really!?" comment is sarcasm. (My apologies if I've misread.) What you're assuming/described is the intuitive situation, and the obvious solution is the correct one. OTOH, the OP described the opposite scenario, and the obvious path of "stable with feet low = raise weight" would be exactly backwards. The horizontal test conveniently resolves that ambiguity.