The dimension conjecture for generalized twisted characters

Let $m\in\mathbb{N}$ and let generalized $\sigma$-twisted characters mean suitably defined generalized characters associated with $\sigma$-twisted modules of the supertriplet vertex operator superalgebra $\mathcal{SW}(m)$.

Generalized twisted-character dimension conjecture. The vector space of suitably defined generalized $\sigma$-twisted characters is $3m+1$-dimensional.

The source presents this as conditional on the existence of $m$ logarithmic modules and on the preceding conjectural dimension formula for the twisted Zhu algebra; the notion of generalized character is not defined there, and no resolution is given.