The rank formula for the even part of the supertrace space of cyclotomic Sergeev algebras

Let ${\rm R}$ be an integral domain in which $2$ is invertible. Let $\mathfrak{H}_{\Delta}(n)^{g}$ be the cyclotomic Sergeev algebra, let ${\rm SupTr}(\mathfrak{H}_{\Delta}(n)^{g})_{\overline{0}}$ denote the even part of its supertrace space, and let $\mathscr{MP}^{\mathsf{0},m}_{n}$ and $\mathscr{MP}^{\mathsf{s},m}_{n}$ be the indexing sets defined in the paper. Rank formula. The ${\rm R}$-module ${\rm SupTr}(\mathfrak{H}_{\Delta}(n)^{g})_{\overline{0}}$ is free, and

$$\operatorname{rank}_{{\rm R}}{\rm SupTr}(\mathfrak{H}_{\Delta}(n)^{g})_{\overline{0}}=\begin{cases} |\mathscr{MP}^{\mathsf{0},m}_{n}|, & \text{if $d$ is even,}\\ |\mathscr{MP}^{\mathsf{s},m}_{n}|, & \text{if $d$ is odd.} \end{cases}$$

This gives the rank of the even supertrace space over an arbitrary integral domain where $2$ is invertible, with the parity of the level determining which indexing set occurs. The supplied text does not establish whether the result is intended as a conjecture or a proved theorem, so its status is left open.