Awesome project! Reminds me of donothingfor2minutes.com from Calm, but with a different end goal of focus instead of calm.
Regarding mobile phones going to sleep, Wake lock [1] might help, unless you can reduce to 59s since I believe 1m is the threshold (make sure to request within the context of the user hitting "start"). Unfortunately on older mobile browsers [2], the best workaround I found was using this NoSleep library[3].
I enjoyed trying to tell if the circle had a 3d aspect to or not, and also trying to tell if it was completely black or actually dark dark gray. I'm not sure if I'm cognitively more focused but I'm going to try it again when I need that focus.