The distance bound for loops with nonequivalent associator

Let $L=(G,\circ)$ and $M=(G,*)$ be Moufang loops on the same set $G$ of order $n$, and define their distance by

$$d(\circ,*)=\left|\{(a,b)\in G\times G: a\circ b\ne a*b\}\right|.$$

Suppose that $L$ and $M$ have nonequivalent associators. Distance-bound conjecture. Their distance satisfies

$$d(\circ,*)\ge 3n^2/8.$$

This conjecture asks for the minimum distance between Moufang loops with nonequivalent associator; the source gives no evidence of a proof or disproof.