The generalized Amitsur exponent conjecture for finite actions

Let $W$ be a unital $F$-algebra over a field of characteristic zero, and let $A$ be a $W$-algebra. Say that the action of $W$ on $A$ is finite when the image of the action has finite dimension. Let $c_n^W(A)$ denote the generalized codimension sequence, and let $\exp(A)$ denote the ordinary PI-exponent of $A$. The generalized Amitsur exponent conjecture. If the action of $W$ on $A$ is finite, then the limit

$$\exp^W(A):=\lim_{n\to \infty}\sqrt[n]{c_n^W(A)}$$

exists and is a nonnegative integer. Furthermore,

$$\exp^W(A)=\exp(A).$$

The ordinary Amitsur conjecture is known in the classical setting, and the source notes equality with the ordinary exponent for the stated Grassmann-algebra examples. The asserted existence and equality for all finite actions remain open.