I've got to agree (1" reel-to-reel, 16-track), but they probably would've scrapped together as much money as possible and gone into a studio, so 32 track is not out of the question.
I have to say 32 tracks is highly unlikely for a couple of reasons. Back then studios had 8 tracks or 16 tracks or 24 tracks. If you wanted more tracks you had to gang together machines and they were almost always of the same format. A studio that could do this (it required a lot of technical stuff to do and in the process meant you lost 2 tracks per deck to get them to sync) usually had enough money for 24 tracks. So if more then 24 tracks were required then you usually ganged two 24 tracks together giving you 44 tracks. Ganging two 16 tracks together gave you only 28 tracks so in most cases you would move to a 24 track machine instead.
Another reason, these studios were expensive. I've been in a lot of LA studios back in the 1980s from some of the biggest to the smallest, and only the big studios were setup to do multiple machines at a time. And these studios where $150+ an hour in 1980s money. Compare that to 16 track studios which cost between $25-50 an hour. And 8 tracks were even cheaper.
Also the cost of tape, 2" reels cost about $100 and you got about 15 minutes of time. Ganging two of these machines together meant 200 bucks ber song just in tape alone, a standard album length project could easly cost $2000 just for the tape. 1" tape was about half the cost and 1/2" was about half of that.