Polynomiality conjecture for medial quasigroups of prime-power order

Let $k$ be any natural number, let $p$ be a prime, and write $mq(\mathbb Z_p^k)$ for the number of medial quasigroup structures on the elementary abelian group $\mathbb Z_p^k$, while $mq(p^k)$ denotes the total number of medial quasigroups of order $p^k$. Polynomiality conjecture. There is an integer polynomial $f_k$ of degree $2k$ such that

$$mq(\mathbb Z_p^k)=f_k(p)$$

for every prime $p$, and there is an integer polynomial $g_k$ of degree $2k$ such that

$$mq(p^k)=g_k(p)$$

for every prime $p$. This proposes uniform polynomial formulas for the two counting functions in each fixed dimension $k$, extending the explicit prime-square calculations and addressing the general counting problem for medial quasigroups of prime-power order.