I should clarify my "at any speed" statement. I didn't clearly state it but I meant that it wobbles at any speed (not that it wobbles at 4x at any speed).

Discovered my rear wheel has a long cut in it. Any deeper and the inner tube would've burst through. I had to replace the wheel. At the same time, I switched back to a different thin wheel (lent the other thin wheel to a friend). I think the bulge in the rear wheel from that cut caused the wobbling. Maybe it wasn't enough to make a noticeable effect with thicker front tires.

Anyway, if it happens again, I'll post an update, but for now, it appears solved. Thanks.