The graded Amitsur conjecture for codimension growth

Let $T$ be a semigroup, let $A$ be a $T$-graded algebra, and let $c^{T\text{-}\mathrm{gr}}_n(A)$ denote its $n$th graded codimension, defined by

$$c^{T\text{-}\mathrm{gr}}_n(A):=\dim\left(\frac{P^{T\text{-}\mathrm{gr}}_n}{P^{T\text{-}\mathrm{gr}}_n\cap \operatorname{Id}^{T\text{-}\mathrm{gr}}(A)}\right).$$

Graded Amitsur conjecture. There exists a graded PI-exponent

$$\operatorname{PIexp}^{T\text{-}\mathrm{gr}}(A):=\lim\limits_{n\to\infty} \sqrt[n]{c^{T\text{-}\mathrm{gr}}_n(A)} \in \mathbb Z_+.$$

This is the graded analogue of Amitsur's conjecture, asserting that the exponential growth rate of graded codimensions exists and is a positive integer. The supplied text does not provide evidence of a resolution.