Aljadeff–author's conjecture on graded Gelfand–Kirillov dimension

Let $G$ be a finite abelian group and let $A$ be a $G$-graded PI-algebra. For each $k\geq 1$, write $\operatorname{GKdim}_k^G(A)$ for the $k$-th $\mathbb{Z}_2$-graded Gelfand–Kirillov dimension and $\operatorname{GKdim}_k(A^{1_G})$ for the corresponding invariant of the identity-component subalgebra $A^{1_G}$. Aljadeff–author's conjecture. For every $k\geq 1$,

$$\operatorname{GKdim}_k^G(A)\leq |G|^2\operatorname{GKdim}_k(A^{1_G}).$$

The conjecture is motivated by the analogous graded PI-exponent inequality and by the positive result for finite abelian groups cited in the paper; the supplied text does not indicate a resolution of this Gelfand–Kirillov-dimension bound.