Possesive pronoun to use with the interrogative pronoun

When using the interrogative pronoun, 'who', what would the possessive form be?

'Who checks X letterbox every day?'.

I feel it ought to be 'his' but some people I know claim it should be 'their', which to me seems to contradict the singular form of the verb 'check'.

'Who checks his letterbox every day?' is what I would say naturally.
'Who checks their letterbox every day?' sounds a bit off to me.

Looking on the internet doesn't really return anything useful, only the use of 'their' as a singular pronoun, which seems to be somewhat popular a topic.

Note, this is specifically regarding the interrogative pronoun; I understand the debate about his/her/their/ones in other circumstances but I want to know whether the same can apply to 'who' or 'whom'.