At first there didn't seem to be any signs that Jun would inherit his spirit except for his having to be Ren's vessel. Because of the corruption and the ceremony that kamio's family did everything got messed up. Since Jun was able to draw all of the clansmen from memory that was the for sure proof that he was probably going to inherit Garai's name. But with his soul laid to rest he may not inherit it..just have the memories. I'm not sure if i answered your question lol
No, it does not seem like he will inherit it. Never did. It has always and does seem to be Hazuki.

especially if the clansmen treat Kouchi as they did Raki...Kamio isn't going to stand for it...I feel like there also may be a way for someone to inherit Garai's name but something drastic and tragic may have to happen..w/ the attitude of those three the clansmen are going to be so over them lol