Frobenius test-module conjecture for finite projective dimension

Let $(R,\mathfrak m,k)$ be a local ring of characteristic $p>0$, let $n>0$, and regard $R$ as the $R$-module ${}^{\varphi^n}\!R$ via the $n$th Frobenius map. Let $M$ be an $R$-module. Frobenius test-module conjecture. If, for some fixed $i,n>0$,

$$\operatorname{Tor}_i^R(M,{}^{\varphi^n}\!R)=0,$$

then $M$ has finite projective dimension. The question is whether the Frobenius-twisted module ${}^{\varphi^n}\!R$ is a test module for finite projective dimension over every local ring; its resolution is left open in the supplied text.