The generalized GWA GK-dimension conjecture

Let $R$ be an affine $\Bbbk$-algebra and let $W:=R(\mathbf x,\mathbf y,\sigma,h)$ be a generalized Weyl algebra of degree $m$. Write

$$G=\langle \sigma_1,\ldots,\sigma_m\rangle.$$

Assume that $R$ is a finitely generated module over the invariant subalgebra $R^G$. GK-dimension conjecture. The equality

$$\operatorname{GKdim} W=\operatorname{GKdim} R+m$$

should hold.