A polynomial bound for multiplication groups of Moufang loops

Let $n$ be the order of a Moufang loop, and let $\operatorname{Mlt}(L)$ denote its multiplication group. Polynomial-bound conjecture. The function

$$f(n)=4n^4$$

solves the problem of finding a function $f:\mathbb N\to\mathbb N$ such that the order of the multiplication group of every Moufang loop of order $n$ is less than $f(n)$. The proposed bound is motivated by the finite Paige loops $M^*(q)$, for which the source derives $|\operatorname{Mlt}(M^*(q))|<4|M^*(q)|^4$; the source does not state whether this claim has been resolved.