McKay–Wanless conjecture on near-quadratic automorphism orders of quasigroups

Let $f(n)$ denote the maximum of the orders of automorphisms of quasigroups of order at most $n$. The preceding discussion gives $f(n)=O(n^2)$. McKay–Wanless conjecture. For every $\epsilon>0$ and for infinitely many values of $n$,

$$f(n)>n^{2-\epsilon}.$$

This conjecture says that the exponent $2$ in the quadratic upper bound is best possible in the sense that values of $f(n)$ come arbitrarily close to $n^2$ along an infinite sequence. The source attributes it to McKay et al.; no resolution is given.