The Huneke–Jorgensen–Wiegand tensor-product Tor-vanishing conjecture

Let $R$ be a complete intersection of codimension $c$, and let $M$ and $N$ be finitely generated $R$-modules. A module is a syzygy if it occurs as a syzygy in a projective resolution. A module $M$ has rank if $M_{\mathfrak p}$ is a free $R_{\mathfrak p}$-module of constant rank at every prime ideal $\mathfrak p$ of $R$.

Huneke–Jorgensen–Wiegand conjecture. If $M\otimes_R N$ is a $(c+1)$st syzygy and $M$ has rank, must

$$\operatorname{Tor}^{R}_{i}(M,N)=0$$

for all $i\geq 1$?

The conjecture asks for conditions forcing the depth formula and total vanishing of positive-degree Tor over complete intersections. It is described in the source as implicit in work of Huneke, Jorgensen, and Wiegand; no resolution is supplied here.