Minimal-dimensional representation conjecture for finite W-superalgebras

Let $\mathfrak{g}=\mathfrak{g}_{\bar 0}+\mathfrak{g}_{\bar 1}$ be a basic Lie superalgebra over $\mathbb{C}$, let $e\in\mathfrak{g}_{\bar 0}$ be nilpotent, and let $\mathsf{r}$ be the associated discriminant number whose parity is the judging parity. Denote by $U(\mathfrak{g},e)$ the finite $W$-superalgebra associated with $(\mathfrak{g},e)$. Minimal-dimensional representation conjecture. If $\mathsf{r}$ is even, then $U(\mathfrak{g},e)$ affords a one-dimensional representation; if $\mathsf{r}$ is odd, then $U(\mathfrak{g},e)$ affords a two-dimensional representation. This conjecture is the finite $W$-superalgebra analogue of Premet's conjecture for finite $W$-algebras. The general case remains open, although special cases were established in the cited work.