Symmetric monoidality conjecture for the Goodwillie derivative functor

Let $C$ be the category appearing above, let $L{\mathsf{Sp}}$ be the corresponding localized category of spectra, and write $\partial^{BB}_\ast$ for the Goodwillie derivative functor on finitary pointed functors. Equip $({\mathsf{Fun}}_\ast^\omega(C,L{\mathsf{Sp}}),\wedge)$ with its indicated symmetric monoidal structure and $({\mathsf{RMod}}_{\partial_\ast {\mathsf{Id}}},\circledast)$ with the indicated monoidal structure. Symmetric monoidality conjecture. The functor

$$\partial^{BB}_\ast: ({\mathsf{Fun}}_\ast^\omega(C,L{\mathsf{Sp}}),\wedge) \longrightarrow ({\mathsf{RMod}}_{\partial_\ast {\mathsf{Id}}},\circledast)$$

can be made symmetric monoidal. This would provide a monoidal comparison between the Goodwillie derivatives and the operadic structure on the derivatives of the identity; the statement is presented as a conjecture in the paper, and its resolution is not established in the supplied text.