Dimension of a graded module over a local$^*$ ring

Question

Assume that $R$ is a positively graded ring which has only one maximal homogeneous ideal $\mathfrak{m}^*$. Let $M$ be a finitely generated positively graded ring over $R$ and consider $\mathcal{l}_R(M/{\mathfrak{m}^*}^n M)$. Where $\mathcal{l}_R$ stands for the length of the $R-$module $M/{\mathfrak{m}^*}^n M$. It can be shown that $\mathcal{l}_R(M/{\mathfrak{m}^*}^n M)$ is a polynomial in $n$ for large $n$. Is the degree of $\mathcal{l}_R(M/{\mathfrak{m}^*}^n M)$ equal to the dimension of $M$?