I just completed 2 self bows with that exact ring pattern. I blew through at least 5 actual rings accidentally before I was able to get to one complete ring on the last one. The first I got lucky with and got the first ring I aimed for. The first bow however, did lift a splinter in the same spot 3 times during tillering. I eventually got mad and decrowned the spot only where the splinter was, then finished my tiller. After I got it to pull all the way back without a splinter, I wrapped the decrowned spot with hemp to add protection just in case. I flattened that area fairly good too.
I would recommend to just say the hell with chasing those rings, and next time I will do this very thing and decrown the entire stave, making the back as wide and flat as you can. No thin ring problems anymore, or bad eye problems with that approach, and you can sinew if you wanna, or leave it as is. Either will work fine. Both of my bows were fairly wide though, about 1.75 inches to be exact. Good luck whatever you decide.