The enumeration conjecture for medial racks and quandles of prime order

Let $p$ be a prime. Write $r_{\emph{med}}(p)$ and $q_{\emph{med}}(p)$ for the numbers of medial racks and medial quandles of order $p$, respectively, and r_{\emph{2-red}}(p) and q_{\emph{2-red}}(p) for the numbers of $2$-reductive racks and $2$-reductive quandles of order $p$, respectively.

Enumeration conjecture.

$$r_{\emph{med}}(p)-r_{\emph{$2$-red}}(p)=p-2$$

and

$$q_{\emph{med}}(p)-q_{\emph{$2$-red}}(p)=p-2.$$

In particular, every medial rack of order $p$ that is not $2$-reductive is a quandle.

This conjecture is supported by the authors' computational data and concerns the enumeration of medial racks and quandles of prime order. Its resolution would also establish the stated structural consequence for medial racks that are not $2$-reductive.