The one-dimensional tensor-product torsion-freeness conjecture

Let $R$ be a one-dimensional local complete intersection domain, and let $M,N\in\operatorname{mod} R$. Tensor-product torsion-freeness conjecture. If $M\otimes_R N$ is torsion-free, then $M$ or $N$ is torsion-free.

This is the one-dimensional specialization of the preceding tensor-product question and remains open in general. The claim follows from a conjecture of Dao asserting, in particular, that all finitely generated modules over one-dimensional local complete intersection domains are Tor-rigid.