Let $\mathbb{A}$ be a polynomial ring in $n$ variables over an algebraically closed field $\mathbb F$. Given a maximal ideal $\mathfrak{m}$ of $\mathbb A$, consider the quotient $\mathbb{A}/\mathfrak{m}$ as a left $\mathbb{A}$-module ($\mathbb{A}/\mathfrak{m}$ is one-dimensional over $\mathbb F$).
Suppose $M$ is a right finitely generated $\mathbb{A}$-module which is infinite dimensional (as $\mathbb F$-vector space) and such that $\dim_{\mathbb F}(M\otimes_{\mathbb{A}} \mathbb{A}/\mathfrak{m})<\infty$ for any maximal ideal $\mathfrak m$.
Question: Can we say that $\dim_{\mathbb F}(M\otimes_{\mathbb{A}} \mathbb{A}/\mathfrak{m})$ is independent of $\mathfrak m$?
The motivation of the question is to show that $M$ is projective (therefore free, by Quillen's Theorem).