The Specht property for finitely generated generalized operator algebras

Let $A$ be a $W$-algebra over a field of characteristic zero. A class of algebras has the Specht property if every $T$-ideal of the class is finitely generated. The Specht conjecture. If $W$ is finitely generated, then $\operatorname{Id}^W(A)$ is finitely generated. Kemer's theorem establishes the analogous property for associative algebras in characteristic zero, while the generalized setting is open; the preceding examples show that the property may fail when $W$ is not finitely generated.