"The iOS version is out! Here's a short FAQ!" post (2 Aug, 2016) on current page 7/19 of the HoN discussions on Steam or Steam website (can't link off-site URL either here or in PM).

Server sync feature—very important to me and rather rare in digital board games —seems to be broken. I just started the U.S. campaign #2. I finished the Oddball's predicament mission on iOS but Oddball is still in need on PC! I am well logged on both sides. Scores are not synchronized either.