I put a small Morena sapodilla at my renters house last spring. I put a shade over it during July, August, and September. It saw temperatures from 30-119 degrees and came out fine with no winter protection, other than a little frost burn on a few leaf tips. It will only get stronger as it gets bigger, so I'm keeping my hopes up.
I also had 4 Haysa sapodillas in 15 gallon containers at my house which is a little dryer, hotter in summer and colder in winter. They seemed happy until temps got over 110 degrees. I put them under 60% shade and they survived, but didn't thrive. In the winter, on an unusually hot day, the plastic I had over them blew down from a gust of wind, and draped over the plants which subsequently fried the top 12 inches of all the trees. But 6 weeks ago, I put them in the ground and all are growing vigorously, one is even flowering. They seem like pretty tough plants.