Premet proved the famous KW-conjecture in modular Lie algebra. After, Premet introduced the finite $W$-algebra. Also, Premet proposed the conjecture every algebra $U(g, e)$ admits a $1$-dimensional representation then, Losev proved this conjecture for g classical.

Anyway, there is an important result from above result, i.e. the reduced enveloping algebra $U_{\chi}(g_{k})$ has a simple module of dimension $p^(dim O(\chi))/2$.

so, a natural question, for the super version, what about these results when we consider the basis classical Lie superalgebra?