I saw a post once about someone equivalating Geiju's personality and not-talking habits to autism. I'm not sure if that was what YanDev was going for, but I think it makes a little sense. I personally don't have much experience with the symptoms of autism, so before I actually make this canon in my story I'd like to do a bit more research.
Outside of that, I'll talk about the relationships I featured up above for Geiju. Shoku, obviously, was a number one choice. We never really see them interact since the two being brothers was unintentional, but regardless, I believe that they have a swell relationship.
The only drama I bothered adding was between him and Miyuji. Do keep in mind that this is completely spontaneous, and I really have no reason for adding any drama between them. So before Miyuji became the iconic, bright-punk 'Miyuji Shan' that we know today, we know that she was far more tame. Simple hair, shy personality, all of that. During that time, she and Geiju were very close friends.
I imagine that outside of Miyuji just being sweet, she got along better with more shy people, and before Geiju became as aloof as he is now, that's what he was. They grew close because Miyuji was a nice person, and because Geiju was easy to get along with if you were patient enough.
If I were to feed into "dating" rumours, I would say that Miyuji's sudden change in fashion and personality repeled Geiju's interest. By all means, he doesn't dislike her, but since she's no longer the Miyuji that he knew in love, he failed to find her presence as pleasant as before. The two still talk and wish each other well, but they're no longer as close as they were before.