The Frobenius operator on q-de Rham complexes

Let $R$ be a smooth $\mathbb Z$-algebra with $p$-adic completion $\widehat R$. The Frobenius conjecture. The $p$-adic completion $\widehat{q\!\operatorname-\!\Omega_R}$ admits a $\mathbb Z[[q-1]]$-semilinear endomorphism of $E_\infty$-algebras

$$\varphi_p:\widehat{q\!\operatorname-\!\Omega_R}\to\widehat{q\!\operatorname-\!\Omega_R},$$

with semilinearity $q\mapsto q^p$. For a framing, it is induced by the Frobenius lift sending $T_i$ to $T_i^p$ and $q$ to $q^p$. Such an operator would supply the Frobenius structure expected from Wach-module theory.