I think you are probably right. Everything I've read gives different answers but I am fairly convinced that there is a connection there. The best explanation I've heard, although of course I don't know if it's true, is called phoral occlusions. Which is caused by sweat and oil clogging the pores. Now, I guess other people's sweat would just make it worse
. But maybe the T-shirt helps because it absorbs the sweat, I don't know.
I've also read something about sweat changing the ph of the skin but that sounds pretty fishy. I have seen all sorts of examples of breakouts happening where you sweat a lot.
Anyway, the only advice I can give is, if you are not already, to take a shower immediately and clean the area as thourougly as you can to wash all the by-products and residue away. If you can't do that then look into getting some wipes you can use to clean up with. Or just a wet towel or cloth would be better than nothing.