Gorenstein and complete-intersection conjecture for modules of derivations

Let $R$ be a local $k$-algebra such that the $R$-module of $k$-derivations $\operatorname{Der}_k(R)$ is finitely generated. Write $\operatorname{Gdim}$ for Gorenstein dimension and $\operatorname{CI\text{-}\dim}$ for complete-intersection dimension. Derivation-dimension conjecture.

If

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

then $R$ is Gorenstein; and if

$$\operatorname{CI\text{-}\dim}(\operatorname{Der}_k(R))<\infty,$$

then $R$ is a complete intersection.

This conjecture connects finiteness of homological dimensions of the module of derivations with the corresponding structural properties of the local algebra. The source presents both implications as an open conjecture.