Product rule for Weiss derivatives

Let $F,G:\mathsf{Vect}_{\mathbb{R}}\to\mathsf{Spec}$ be functors, and let $\partial_\ast F$ denote the Weiss derivatives, regarded as right $K(\mathsf{OEpi})$-modules.

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

$$\partial_\ast(F\wedge G)\simeq\partial_\ast F\circledast\partial_\ast G.$$

This is the product rule expected to follow from the conjectured Day-convolution monoidality for $\mathsf{OEpi}$. It remains conjectural in the paper.