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.

so, a natural question, for the super version, what about these results when we consider the basis classical Lie superalgebra,i.e, whether every super W-algebra admits a $1$-dim rep??