The Frobenius–Witt cotangent complex over the field with one element

Let $R$ be an animated $Z_{(p)}$-algebra, and write $R/^Lp$ for its mod-$p$ reduction. The Frobenius–Witt cotangent complex $F\mathbb L_R$ is defined over $\mathbb F_p$, and $F$ denotes the Frobenius endomorphism of $R/^Lp$.

Conjecture on the cotangent complex over the field of one element. There exists a functor

$$\mathbb L_{(-)/\mathbb F_1}:\operatorname{Ani(Ring)}_{\mathbb Z_{(p)}/}\to \mathcal{D}(\mathbb Z_{(p)})$$

of $\infty$-categories, together with a natural equivalence

$$F^*(\mathbb L_{R/\mathbb F_1}\otimes_R^L R/^Lp)\simeq F\mathbb L_R.$$

This conjecture would extend the Frobenius–Witt cotangent complex from a complex over $\mathbb F_p$ to a cotangent complex over $\mathbb Z_{(p)}$, assuming the existence of the field with one element. It is motivated by the Frobenius pullback of the modulo-$p$ fundamental fiber sequence and the expected transitive fiber sequence for a conjectural map $\mathbb F_1\to R\to S$.