I'm pretty sure there's an O-ring inside of the die that holds the seater stem from moving around. I would try and match it up at the hardware store, with the same size or maybe a little thicker. Clean out the die with alcohol and put the new O-ring in without any oil or anything. Maybe the a fresh O-ring will hold things in place better. I'm not guaranteeing anything, but it'd be a cheap thing to try.
ETA: DOC beat me by a minute with a similar response!