Tor-rigidity conjecture for local complete intersections

Let $R$ be a local complete intersection of arbitrary dimension. Let $M$ and $N$ be $R$-modules such that $M$ is locally free of constant rank on $U_R=\operatorname{Spec}(R)-\{m\}$ and $[N]=0$ in the reduced Grothendieck group $\overline G(R)_{\mathbb Q}$ with rational coefficients.

Tor-rigidity conjecture. The pair $(M,N)$ is $\operatorname{Tor}$-rigid: for any $i>0$, the condition $\operatorname{Tor}_i^R(M,N)=0$ forces $\operatorname{Tor}_j^R(M,N)=0$ for every $j\geq i$.

The source states that this is known for regular rings and for hypersurfaces in equicharacteristic or unramified regular local rings, but remains open in general. In particular, the finite-length case is identified as a simple unknown case.