Self-converse classification conjecture for entropic Mendelsohn triple systems

Let $M$ be a finite abelian group, and let $\emph{Lin}(M,R)$ be a Mendelsohn triple system. A Mendelsohn triple system is self-converse when it is isomorphic to its converse, obtained by reversing the directed triples.

Self-converse classification conjecture. The system $\emph{Lin}(M,R)$ is self-converse if and only if no prime dividing the order of $M$ is congruent to $1$ modulo $3$.

This conjecture seeks a complete characterization of self-converse entropic Mendelsohn triple systems. The corresponding statement for DNR Mendelsohn triple systems is proved in the source; the general entropic case is presented as conjectural and is implied by the preceding odd-power classification conjecture.