Herzog–Vasconcelos conjecture on derivation modules

Let $R$ be a Noetherian local ring with residue field $k$, and let $\operatorname{Der}_k(R)$ denote the $R$-module of $k$-derivations of $R$. Herzog–Vasconcelos conjecture. If

$$\operatorname{pd}_R\operatorname{Der}_k(R)<\infty,$$

then $\operatorname{Der}_k(R)$ is a free $R$-module. This is introduced as an open problem related to the Zariski–Lipman conjecture; the paper does not report a general resolution.