Gabber-type maximal Cohen–Macaulay conjecture for reflexive modules

Let $R$ be a local complete intersection of dimension $3$. Let $U_R=\operatorname{Spec}(R)-\{m\}$ be the punctured spectrum, and let $N$ be a reflexive $R$-module that is locally free of constant rank on $U_R$. Write $\overline G(R)_{\mathbb Q}$ for the reduced Grothendieck group of $R$ with rational coefficients.

Generalized Gabber conjecture. If $[N]=0$ in $\overline G(R)_{\mathbb Q}$, then $\operatorname{Hom}_R(N,N)$ is a maximal Cohen–Macaulay $R$-module if and only if $N$ is free.

This is presented as a possibly weaker version of the question whose affirmative resolution would imply Gabber's conjecture. The source discusses it as open.