Herzog's cotangent homology conjecture

Let $R$ be a regular local ring and set $S=R/I$. Let $\operatorname{T}_i(S/R,S)$ denote the cotangent homology modules.

Herzog's cotangent homology conjecture. The following conditions are equivalent:

1. $I$ is a complete intersection.
2. $\operatorname{T}_i(S/R,S)=0$ for $i>1$.
3. There exists an integer $i_0>1$ such that $\operatorname{T}_i(S/R,S)=0$ for $i\geq i_0$.

The conjecture concerns whether eventual vanishing of cotangent homology detects complete intersections. The paper's appendix states that Herzog's conjecture has been settled in some cases, including ideals in the linkage class of a complete intersection, but remains wide open in general; the supplied resolved status for this formulation should therefore be checked against the cited development.