I love facts and try to avoid too much speculation, but I would bet money on OH being a fairly dull bald/balding man.
My evidence for this is that Jack tends to adopt her partners' personalities as her own (Foodie Jack in the Allegra period, Veggie Jack during LJC's time), and the only thing that has really changed about her recently is a love of hats. Past a certain age, men with a full head of hair tend to go hatless. Therefore, the Onion Police should be looking for a meat-eating bald man with no obvious personality traits...
https://giphy.com/d3mlE7uhX8KFgEmY
ETA: to add, on a detective note, Jack put a poll on Twitter about what to cook for a special sexy meal on 8th Jan. This could have been the start of it all...