Soo... This stack is an exterior stack that terminates 2' above the peak? It terminates therefor at least 2' above the HIGHEST portion of the home, not just where it exits? IF NOT then that is the problem, the chimney was never built properly; even it it 2' above the highest portion of the home, that's not tall enough, a minimum of 3' is barely adequate, and 4' is minimum recommended.

Having said all that, What "chimney professional" talked you into chimney caps? What "problem" were you trying to solve with them? Have them REMOVED and solve your problem. It's likely that the second flue always had a week and occasionally reversing draft due to improper chimney height, but the caps (which are, in most cases, unnecessary) have caused this unseen problem to become visible. You have three options: 1. extend the chimney height so that it functions like a proper chimney should. 2. Remove the caps! (you have some aversion to this, but it works and will cost you less than the other options). Remove furnace connect from basement flue and disable furnace (not very likely a good choice).