Mars has its own "ice ages"
Pretty Much the same amount of ice is always there, just, as the orbit changes in obliquity and spin-axis inclination, the ice moves from the Poles to the equator and back again.
it's a cycle that repeats.
Whether ice melts or not, is a more difficult question. Sublimation is fast enough to remove all that ice, and the mere presence of ice is enough to buffer temperatures to prevent melting except in extreme circumstances.
Jim Head's a nice guy, but he and his students are very watery in their thoughts, imho...