Odd-power Eisenstein module classification and enumeration of linear Mendelsohn triple systems

Let $k\geq 3$ and let $\emph{Lin}(\mathbb{Z}[\zeta]/(1+\zeta)^{2k+1},R)$ be a Mendelsohn triple system, where $\zeta$ denotes a primitive cube root of unity. Set

$$M=\mathbb{Z}/_{3^k}\oplus\mathbb{Z}/_{3^{k+1}}.$$

The notation $T^\psi$ refers to the endomorphism defined from the ring homomorphism $\psi$ as in the preceding small-order classification.

Odd-power classification conjecture. There is a quasigroup isomorphism

$$\emph{Lin}(\mathbb{Z}[\zeta]/(1+\zeta)^{2k+1},R)\cong \emph{Lin}(M,T^\psi).$$

Moreover, the number of linear Mendelsohn triple systems of order $3^n$ is

$$l(3^n)=P(n),$$

where $P(n)$ denotes the partition function.

This conjecture extrapolates the classifications established for the first few odd powers of $1+\zeta$ and predicts that the enumeration of linear Mendelsohn triple systems of order $3^n$ is governed by integer partitions. Its status is unresolved in the supplied source.