That seems really paradoxical and I think it would just burn up compute. The AI really doesn't have any way to know it's getting better without humans telling. As soon as the AI begins to recursively improve based on its own definition of improvement model collapse seems unavoidable.

But if the second AI that can self improve comes up?

Then it all remains a question of who has the most compute power, as self improve seems compute heavy with the current approach.

If that happens catching up will be meaningless, everything we know and care about will change. You don’t have to be doomsday about it even, a self improving AI will quickly be more efficient than a human brain, all the data centers will be useless, tech companies will collapse (so will most others), everyone will have an incredible AI resource for the price of a hotdog. There’s no way it wouldn’t leak from whoever made it, either by people or by the AI itself.