I don't think progressive fancy gender-ambiguous pronouns such as "ze" and "zir" are required in this case.
OP's profile says her gender is female. That's one. "Primary partner" is referred to as "his", so that's a guy. OSO is referred to as "she" and metamour is referred to as "her", but we are unclear as to whether or not these two are the same person. There is only one other party in the "sort-of quad" whose sex and gender are not specified. It's probably a guy, as this would lend to the logic of it being two male/female couples in said "sort-of quad".