Sofic groups and finite Moufang loops

Let $G$ be a group. A group is sofic if, for every finite set $X\subseteq G$ and every $\varepsilon>0$, there is a finite quasigroup $(Q,*)$ containing $X$ such that multiplication agrees on products in $X$, the unit law fails on fewer than an $\varepsilon$-fraction of elements, and associativity fails on fewer than an $\varepsilon$-fraction of elements for each relevant pair. A group is locally embeddable into finite Moufang loops if every finite subset of $G$ admits an injective map into a finite Moufang loop preserving all products that remain in the subset.

Sofic–Moufang conjecture. A group $G$ is sofic if and only if it is locally embeddable into finite Moufang loops.

The conjecture proposes that finite Moufang loops provide the appropriate associative-enough subclass of finite IP loops for characterizing sofic groups. The source presents it as the second, less probable alternative to the preceding conjecture; its resolution is not supplied.