Mixed chain rule for Weiss derivatives

Let $F:\mathsf{Vect}_{\mathbb{R}}\to\mathsf{Spec}$ and $G:\mathsf{Spec}\to\mathsf{Spec}$ be functors, with $F(\mathbb{R}^{\infty})=\ast$. Let $\partial_\ast$ denote Weiss derivatives and let $\odot$ be the composition product of symmetric sequences and right modules.

Mixed chain-rule conjecture. There is an equivalence of right $K(\mathsf{OEpi})$-modules

$$\partial_\ast(G\circ F)\simeq\partial_\ast G\odot\partial_\ast F.$$

This reformulates the chain rule proposed by Arone and Ching in the language of right $K(\mathsf{OEpi})$-modules. The result is not proved here.