Enumeration of simple optiongraphs

Determine whether there is a reasonable enumeration of simple optiongraphs with n positions.

Background

An optiongraph is simple when its maximum congruence relation is trivial. The paper gives counts of simple optiongraphs for small numbers of positions—2 with one position, 3 with two positions, 15 with three positions, and 250 with four positions—and displays the cyclic examples with three positions. It leaves unresolved whether these data admit a useful enumeration for arbitrary n.

References

Is there a reasonable enumeration of simple optiongraphs with $n$ positions?

— Isomorphism Theorems for Impartial Combinatorial Games  (2505.04919 - Baltushkin et al., 8 May 2025) in Question in Section 6.2, “Simplicity”