Papadimitriou–Yannakakis' completeness conjecture for minimum generating sets of quasigroups

A quasigroup is an algebra whose multiplication table is a Latin square; equivalently, for all elements $a,b$, there are unique elements $x,y$ such that $ax=b$ and $ya=b$. The minimum generating set problem asks whether a given quasigroup has a generating set of size at most $\log n$, where $n$ is the order of the quasigroup.

Papadimitriou–Yannakakis' conjecture. The minimum generating set problem for quasigroups is $\exists^{\log^2 n}\mathsf{P}$-complete.

Papadimitriou and Yannakakis established the analogous completeness result for arbitrary magmas and conjectured that completeness persists for the more structured class of quasigroups. The paper reports an unconditional refutation of a version of this conjecture for completeness under quasi-polynomial-size constant-depth reductions, and a refutation under polylogarithmic-space reductions assuming $\mathsf{EXP}\neq\mathsf{PSPACE}$, so the conjecture is treated as refuted.