Fortunately, my colleague knew that there is political science to help us:

This study probes one particular component of the well documented linkage between personal appearance and impression formation by investigating the extent to which and the mechanisms through which bald and balding men are underrepresented in high elective office. Study 1 compares the prevalence of hair loss among governors and members of Congress, on the one hand, and the general public, on the other, and concludes that officeholders are much more likely to have a full head of hair than would be expected of men of their age. Study 2 poses an experimental test of voter bias against bald and balding candidates by presenting voters in a simulated congressional race with materials depicting otherwise identical candidates in either their natural bald or balding condition or wearing a professionally fitted hairpiece. No voter bias against bald or balding candidates is apparent, a finding that suggests that the causal mechanism underlying underrepresentation of bald and balding men is not voter bias.