This person" is singular so the correct word is his. Singular possessive. Case closed.
"Their, them, themselves, they (as singular pronouns)
Many English speakers believe that using the plural pronouns they, them, themselves, and their in gender-neutral singular constructions is incorrect. For example, these people would consider the them in “call a friend and ask them to come over” to be wrong because them by definition refers to multiple people, whereas in this clause its antecedent (a friend) is singular.
But there are problems with this view."
You are inferring a gender when it hasn't been provided to you. Keep on swinging and missing.