G-dimension and complete-intersection-dimension conjecture for derivation modules

Let $R$ be the local ring considered in the source, and write $\operatorname{Der}(R)$ for its module of derivations. Let $\operatorname{G-dim}_R$ and $\operatorname{CI-dim}_R$ denote Gorenstein dimension and complete-intersection dimension, respectively.

Derivation-module dimension conjecture. If

$$\operatorname{G-dim}_R\operatorname{Der}(R)<\infty,$$

then $R$ is Gorenstein; respectively, if

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

then $R$ is a complete intersection ring.

The source states that the G-dimension case has been settled affirmatively for graded Cohen–Macaulay rings of minimal multiplicity, while the full conjecture remains unresolved in the stated generality.