Frobenius powers of non-complete intersections

For a commutative ring $R$ of characteristic $p$, let $\varphi :R\to R$ be the Frobenius homomorphism and let ${}^{{\varphi }^r}R$ denote the $R$-module structure on $R$ defined via the $r$-th power of the Frobenius. We show that the Tor functor against the Frobenius module, ${\text{Tor}}_{\ast }^R(-,{}^{{\varphi }^r}R)$, is rigid for a certain class of depth zero rings which includes rings that are not complete intersection. We also show that ${\text{Tor}}_{\ast }^R(-,{}^{{\varphi }^r}R)$ is not rigid (non-vacuously) when $\text{depth}(R)>0$ and $r$ is large enough. This answers a question of Avramov and Miller: does rigidity of ${\text{Tor}}_{\ast }^R(-,{}^{{\varphi }^r}R)$ hold for non-complete intersections?