When is Hilbert-Samuel multiplicity of a local ring non-increasing along localization at prime ideals?

Question

For Noetherian local ring $(R,\mathfrak m)$, let $e(R)$ denote the Hilbert-Samuel multiplicity of $R$ with respect to $\mathfrak m$ (https://en.m.wikipedia.org/wiki/Hilbert%E2%80%93Samuel_function#Multiplicity. https://stacks.math.columbia.edu/tag/0AZU).

My question is: If $R$ is a Gorenstein local ring, then is it true that $e(R_{\mathfrak p})\le e(R)$ for every prime ideal $\mathfrak p$ of $R$?

Answer 1

This is a theorem of Nagata and is true for even Cohen Macaulay rings.

(Reference: Nagata 'Local Rings' Theorem 40.1, p.153)