Ding's index–generalized Löewy length conjecture for Gorenstein local rings

Let $(R,\mathfrak{m},k)$ be a Gorenstein local ring of dimension $d$. The Ding conjecture.

$$\operatorname{index}(R)=\operatorname{g\ell\ell}(R).$$

This conjecture concerns the relationship between Auslander's index and generalized Löewy length. It is refuted by examples of one-dimensional local complete intersections of embedding dimension three with index $5$ and generalized Löewy length $6$.