Asymptotic dimension conjecture for diagonal coinvariant rings of monomial groups

Let $\bell$ and $m$ be positive integers with $\bell ygeq 2$ and $m$ dividing $\bell$, and let $G(\bell,m,n)$ denote the corresponding monomial group. Let $R_{G(\bell,m,n)}$ be its diagonal coinvariant ring. Asymptotic dimension conjecture.

$$\lim_{n \to \infty} \frac{\operatorname{dim}(R_{G(\ell,m,n)})}{(\ell (n-1) + 2\ell/m-1)^n}=1.$$

This proposes that the displayed lower bound gives the principal asymptotic term for the dimension of the diagonal coinvariant ring as $n$ tends to infinity; the supplied text gives no resolution.