Local embeddability of groups into finite Moufang loops

Let $G$ be a group. A group is locally embeddable into a class of loops if every finite subset of $G$ admits an injective map into a loop in that class preserving all products that remain in the finite subset. A Moufang loop is a loop satisfying

$$x(y(xz))=((xy)x)z.$$

Local embeddability conjecture. Every group is locally embeddable into finite Moufang loops.

This is the first of two proposed responses to the problem of characterizing groups locally embeddable into finite Moufang loops, and the authors regard it as more probable. It would follow from an affirmative answer to the question of whether the class of all Moufang loops has the Finite Embedding Property.