It is one of those self-fulfilling prophecies.. you test people's T level, and look at their rates of sickness..
By doing that test your are infering a correlation between cause (T-level) and effect (sickness levels) without having first proved that a viable link exists..
And any study done like that would show, those with higher T levels (who would be generally healthier anyway) would have less sickness... For the same reasoning, I would doubt a peer reviewed scientific paper would show the opposite either.. The basis of testing is flawed at the onset..
In the paper
Associations between male testosterone and immune function in a pathogenically stressed forager-horticultural populationBenjamin C Trumble, (cannot link not allowed)..
It discusses
..Endogenous testosterone appears to be immunomodulatory rather than immunosuppressive. Potentially costlier forms of immune activation like those induced by PHA (largely T-cell biased immune activation) are down-regulated in men with higher testosterone, but testosterone has less impact on potentially less costly immune activation following LPS stimulation (largely B-cell mediated immunity).
WhichI roughly translate into, it does not have much downward effect that is damaging and it is reproductively advantageous in the population overall.. But then I am no doctor... so best ask your endocrinologist.....