He spent the first three loops being a good boy and going to class (to check and make sure that everything really WAS the same from loop to loop, you know), so he does recognize her! And he's going to attempt to get her attention with a soft "excuse me!" because he wants to ask her some questions about the mirror. He'll offer to help her move the mirror, and he'll be making use of inquire while questioning her. (and his talent again, if that's applicable here)

If she'll consent to be questioned, he'll introduce himself (as Shuichi Saihara, a Chemistry student, because what is using a fake identity?) and then ask her what the mirror is and what its purpose is. He can tell that it's magical, but since he only sees his own reflection and nothing about it physically seems off, he's both confused and curious about what it does. He'll also ask if she knows (or at least has a guess about) who put it out here in the first place, why they would do so, and why this spot in particular. He's also curious as to why it's valuable (he's sorry he couldn't help but overhear her mention that). If she doesn't accept his offer to help move the mirror, he'll also ask her where it's SUPPOSED to be, if not here.

If she doesn't want to answer his questions or tries to brush him off, he'll wait until the next loop and try again, this time using friends on her before approaching to try and get the information.