Finite-dimensional representability conjecture for generalized polynomial identities

Let $A$ be a finitely generated $W$-(super)algebra.

Finite-dimensional representability conjecture. There exists a finite-dimensional $W$-(super)algebra $B$ such that

$$\operatorname{gId}(A)=\operatorname{gId}(B).$$

This conjecture would provide the key step toward a generalized representability theorem for varieties of $W$-algebras: together with the preceding results, it would imply that the generalized polynomial identities of every finitely generated $W$-(super)algebra are realized by a finite-dimensional one.