Premet proved the famous KW-conjecture in modular Lie algebra. After, Premet introduced the finite $W$-algebra $U(g, e)$. 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??
EDIT: Thanks for Professor José Figueroa-O'Farrill pointing out that in the literature of mathematical physics, the finite W-algebras appeared in the work of de Boer and Tjin from the viewpoint of BRST quantum hamiltonian reduction.