Isn't testosterone also an immunosuppressant? It would need to be a pretty big effect to merit the tradeoff.

Maybe. From Sullivan:

For reasons no one seems to understand, testosterone may also be an immune suppressant. High levels of it can correspond, as recent studies have shown, not only with baldness but also with heart disease and a greater susceptibility to infectious diseases. Higher levels of prostate cancer among blacks, some researchers believe, may well be related to blacks' higher testosterone levels. The aggression it can foster and the risks it encourages lead men into situations that often wound or kill them. And higher levels of testosterone-driven promiscuity make men more prone to sexually transmitted diseases. This is one reason that men live shorter lives on average than women. There is something, in other words, tragic about testosterone. It can lead to a certain kind of male glory; it may lead to valor or boldness or impulsive romanticism. But it also presages a uniquely male kind of doom. The cockerel with the brightest comb is often the most attractive and the most testosteroned, but it is also the most vulnerable to parasites. It is as if it has sacrificed quantity of life for intensity of experience, and this trade-off is a deeply male one.