No - THEIR - not in gender but in people, plural.
It’s proper English usage.
They don’t have </i>a</i> back. You are trying to remove gender from the language because ultimately it offends the sensibilities of the leftist professors who insist there is no distinction between the sexes so there must be no distinction among the genders. When you give them the language you give them the tools with which you think and will accept progressively more of their assertions because you cannot speak or think of them in any other way.