Let $k$ be a field and $R$ a $d$-dimensional integral domain over $k$. Noether normalization produces an injective algebra homomorphism $k[x_1,\dots, x_d] \to R$ which is module-finite.

Given a maximal ideal $\mathfrak{m} \in \mathrm{Spec}(R)$, can one always find a Noether normalization such that $R_{\mathfrak{m}}$ is flat (= free) over $k[x_1,\dots,x_d]_{\mathfrak{m}^c}$?