Bimodule-valued mixed chain rule for Weiss derivatives

Let $F:\mathsf{Vect}_{\mathbb{R}}\to\operatorname{Top}_\ast$ be a functor, and let $\partial_\ast(\operatorname{Id})$ denote the derivatives of the identity functor on pointed spaces. The derivatives $\partial_\ast F$ are expected to carry a bimodule structure over $\partial_\ast(\operatorname{Id})$ and $K(\mathsf{OEpi})$.

Bimodule chain-rule conjecture. For functors

$$\mathsf{Vect}_{\mathbb{R}}\xrightarrow{F}\operatorname{Top}_\ast\xrightarrow{G}\operatorname{Top}_\ast,$$

with $F(\mathbb{R}^{\infty})=\ast$, there is an equivalence of $(\partial_\ast(\operatorname{Id})-K(\mathsf{OEpi}))$-bimodules

$$\partial_\ast(G\circ F)\simeq\partial_\ast(G)\odot_{\partial_\ast(\operatorname{Id})}\partial_\ast(F).$$

This is the expected unstable generalization of the mixed chain rule, analogous to the bimodule structure on Goodwillie derivatives. The paper presents it as an expectation and does not prove it.