[Oh the door is absolutely not locked. He probably isn't even allowed to lock it after he repeatedly needed to be fetched for missions and bodily dragged out of bed due to hangover. Probably that could be waived now that McGillis is around though.

That said, while Izou certainly doesn't want them being walked in on, he also clearly has no issue with people surmising they're fucking, case in point as he frowns at McGillis in mild confusion;]

Why the hell would ya need to make up a lie like that?

[A beat.]

You embarrassed fer people to think we're fuckin' or something? What, you worried people won't take you seriously if they know yer foolin' around with a dog like me, huh?
[Oh no, his insecurities.]