Finitely generated modules over Noetherian local ring that become isomorphic after faithfully flat base change

Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $S$ be a Noetherian ring which is a faithfully flat $R$-algebra. If $M,N$ are finitely generated $R$-modules such that $M\otimes_R S \cong N \otimes_R S$ (isomorphic as $S$-modules), then is it true that $M\cong N$?

I know this is true when $S=\widehat R$ is the $\mathfrak m$-adic completion of $R$, and in that case it follows from a result of Guralnick which says that if $M/\mathfrak m^n M \cong N/\mathfrak m^n N$ for all $n\gg 0$, then $M\cong N$.

Is the claim indeed true for any Noetherian faithfully flat $R$-algebra? If this is known, what is a good reference?