Herzog's last theorem

Let $R$ be a regular local ring, let $I\subset R$ be a proper ideal, and set $S=R/I$. For an $R$-algebra homomorphism $R\to S$ and an $S$-module $M$, write $T_i(S/R,M)$ for the cotangent modules.

Herzog's last theorem. The following conditions are equivalent:

1. $I$ is a complete intersection.
2. $T_i(S/R,S)=0$ for all $i>1$.
3. $T_i(S/R,S)=0$ for all $i\gg1$.

Herzog posed this conjecture in 1981, and the paper states that it remains open. It predicts that the vanishing of the cotangent modules in all sufficiently high degrees, or in every degree greater than one, characterizes complete intersections in regular local rings.