Bebeacua's multiplicity-two characteristic-sequence conjecture

For a partition $\lambda$ of $n$, let $\mathfrak{C}(\lambda)$ denote the characteristic sequences having shape $\lambda$. Let $\mathfrak{S}_n^3$ denote the set of score sequences of $3$-tournaments on $Z_n$.

Bebeacua's conjecture. For every positive integer $n$,

$$|\mathfrak{C}(2^n)| = |\mathfrak{S}_n^3|.$$

This conjecture is based on matching values for small $n$ and is stated as implying later conjectures in the paper; the source gives no proof or resolution.