Maybe instead they'll be like Feigenbaum constants: apparently universal constants that we don't really understand. For instance, we don't even know for sure that Feigenbaum constants are irrational, though mathematicians who study them strongly suspect they're transcendental.

https://en.wikipedia.org/wiki/Feigenbaum_constants

I feel it’s more likely there is an n^2 algorithm using some radically different approach.

I would assume n^(2+epsilon) where epsilon can be made arbitrarily small, but must be positive.