Strong Zariski–Lipman conjecture

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 $R$. Strong Zariski–Lipman conjecture. If

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

then $R$ is regular. This is presented as a question introduced by Ludington and remains open according to the supplied status.