The minimal-dimension conjecture for finite W-superalgebras

Let $\mathfrak{g}$ be a basic Lie superalgebra over $\mathbb{C}$, let $e\in\mathfrak{g}_{\bar 0}$ be nilpotent, and set $d_1=\dim\mathfrak{g}_{\bar 1}-\dim\mathfrak{g}^e_{\bar 1}$. Let $U(\mathfrak{g},e)$ denote the finite $W$-superalgebra associated with $(\mathfrak{g},e)$. Minimal-dimension conjecture. If $d_1$ is even, $U(\mathfrak{g},e)$ affords a one-dimensional representation; if $d_1$ is odd, it affords a two-dimensional representation of minimal dimension. This conjecture extends Premet's conjecture for finite $W$-algebras; the even case is established for $\mathfrak{gl}(M|N)$ and $\mathfrak{sl}(M|N)$, while the general assertion remains open.