Let $(A, \mathfrak{m}, k)$ be a local ring, and let $\mathrm{gr}_{\mathfrak{m}}(A)$ be its associated graded ring. On page 123 of Atiyah-MacDonald, the following is stated without proof, and not given as an exercise:
It can also be shown that if $A$ is a local ring and $\mathrm{gr}_{\mathfrak{m}}(A)$ is an integrally closed integral domain, then $A$ is integrally closed.
Restricting to the case of regular local rings, i.e. $\mathrm{gr}_{\mathfrak{m}}(A) \cong k[T_1, T_2, \ldots, T_d]$, this clearly follows from the theorem of Auslander and Buchsbaum that $A$ is a unique factorization domain. Is this true for reasons independent their result (or the global dimension argument by Serre)?