Its working name is ‘super tic-tac-Go’, based on the mechanics of ‘Super tic-tac-toe’, which is played on a 9x9 grid consisting of 9 3x3 grids:
Each move you place on a 3x3 grid dictates which one of the nine 3x3 grids (nonant? enneant?) the next player may play in. So if, for example, I play in the top-left corner of the lower-right nonant, you’d next play anywhere in the upper-left nonant.
Some friends and I developed the Go version of this while we were living at Devin’s Baduk House in Ohio, and in our playtesting, determined that self-capture should be allowed, and passing stones should be required. You can pass anytime, but this allows your opponent to play in any nonant. The 3x3 boards are all connected, so a single chain can share liberties between multiple nonants.
You can see two examples of games that I played here on OGS: one with self-capture allowed, and one without.
It’s possible for ko to happen, but it’s so incredibly rare that I don’t think it’s even necessary to have a ko rule. I’d like to do substantially more playtesting to determine the optimal ruleset; also, it may require additional restrictions for the first few moves, as it’s possible for the second player to force the first player to play their first 9 moves in the same nonant. I’m not entirely sure this is an optimal strategy for the second player, though, so it might not matter.