The center-and-trace dimension conjecture for twisted and untwisted Zhu algebras

Let $m\geq 1$ and let $\mathcal{SW}(m)$ be the supertriplet vertex operator superalgebra. Let $A(\mathcal{SW}(m))$ and $A_{\sigma}(\mathcal{SW}(m))$ denote its untwisted and $\sigma$-twisted Zhu algebras. For an associative algebra $A$, let $Z(A)$ denote its center and let $T(A)=A/[A,A]$ denote its trace group.

Center-and-trace dimension conjecture.

$$\dim Z(A(\mathcal{SW}(m)))+\dim Z(A_{\sigma}(\mathcal{SW}(m)))=9m+3,$$

and

$$\dim T(A(\mathcal{SW}(m)))+\dim T(A_{\sigma}(\mathcal{SW}(m)))=9m+3.$$

This conjecture relates the centers and trace groups of the untwisted and twisted Zhu algebras to the modular dimension of the character space; its status is not resolved in the source.