The asymptotic upper-bound conjecture for medial quandles

A medial quandle is a quandle satisfying $(x*y)*(u*v)=(x*u)*(y*v)$ for all elements $x,y,u,v$. Let $n$ be a positive integer and consider medial quandles of order $n$.

Asymptotic upper-bound conjecture. The number of isomorphism classes of medial quandles of order $n$ is at most

$$2^{(\frac14+o(1))n^2}.$$

The paper has established the weaker upper bound $2^{(\frac12+o(1))n^2}$, improving Blackburn's general-quandle bound, while computational results suggest that the exponent $\frac14$ is attainable as an upper bound. No resolution is given here.