In this answer, André Nicolas proves that it is rare for a binary operation on a finite set to be associative, in the following sense: if $A_n$ denotes the number of semigroups that can be defined on a set with $n$ elements, and $B_n$ denotes the number of magmas that can be defined on a set with $n$ elements, then $A_n/B_n\to0$ as $n\to\infty$. (NB for the purposes of this question, two magmas are considered to be the same if they are set-theoretically equal; simply being isomorphic or anti-isomorphic is not enough.)
In the original question, it is asked whether the sequence $(A_n/B_n)_{n\in\mathbb N}$ is monotonically decreasing, but as far as I can tell, this has not been answered. Is this the case?