I was watching some old footage of the US Fest from 1983 and noticed Warren Cuccurullo playing a 2x4 style guitar. At first I thought it was a La Baye until I got a better look.
Here's a blurb on the La Baye for anyone interested:
Here the best video of the guitar I've found. If Dale Bozzio's shrieking bothers you, just turn down the sound. The guitar has to compete for face time with Dale's fishbowl bra but you can get some good looks from time to time. I still like Terry Bozzio's drumming.
When I finally saw the Vox style headstock, I thought that it was a Jennings Winchester guitar.
Based on a thread I found on the Gear Page, it is apparently a custom guitar built by Frank Zappa's guitar tech: