I believe that he's just afraid that he might misplace it or what that's why he's not wearing it. :)
I'm more annoyed that after all these years Nemu still says to his friends he has a "girlfriend" -_-
You wrong, Nemu not the only one but Maya also, Maya also didn't denied when his friend said that he got a gf, the same as Nemu, the one who told Nemu friend he got a gf was Okino, he could just said lover, tho both of them told their friends they already got a lover just didn't tell other their gender.
Ya, I think you were right, he likes the ring just afraid he will lose it or something, plus he didn't even told his mom yet since the reason never come out maybe in the near future chapter he will wear it so his mom will ask him and then his will tell his mom that he got a bf
It still kind of annoys me that Nemu won't wear the ring. Or should I say, I still don't understand why he wouldn't want to? The guy you love gave it to you and you accepted it. I mean, it's just a ring and a symbol but still... why not wear it? I'd be insulted tbh.