I think he has a strong moral compass. He doesn’t like any bully picking on the weak, but especially not one he has somewhat formed a bond with. I can see the story ending where there is a big dramatic falling out, or Anya’s story is solved but the lie is left ongoing for humor.
Well i just wanna say that from my pov Loid genuinely likes Anya at this point and he feels very comfortable with yor, his mission comes first obviously but it all depends on author, i would be very surprised if any dramatic falling happens between Loid and Yor. Both of their secrets will even out each other's.
Now you got me thinking, answer me please if you remember Was it revealed how Yor parents died? Well even if they got killed in war, Loid is atleast mature enough to see that it wasn't Yor fault that his parents died and vice-versa for Yor too. The only problem is the little bro lol but he will come around, afterall he loves his sister.
Is it really just a facade or does Loid truly care about Yor? Whatever the answer but I know he's gentlemen enough to do that.