Symmetric monoidality of the Blans–Blom Goodwillie derivative functor

Establish that the Blans–Blom Goodwillie derivative functor ∂_*^{BB}: (Fun^ω(C, LSp), ∧) → (RMod_{∂_* Id_C}, ⊛) admits a symmetric monoidal structure with respect to the pointwise smash product on functors and Day convolution on right modules, for any compactly generated, differentially dualizable ∞-category C with Sp(C) ≃ LSp.

Background

The paper discusses two constructions of operadic right-module structures on Goodwillie derivatives: one via the coendomorphism operad K(CoEnd(Σ^∞_C)) developed in this work (with a proven product rule), and another via the monoidal derivative functor under composition from Blans–Blom (denoted ∂*^{BB}). To align these approaches, the authors point out that having ∂*^{BB} be symmetric monoidal with respect to the smash-product/Day-convolution structure would parallel the product rule established for their construction.

The conjecture addresses upgrading the known monoidality of ∂_*^{BB} at the level of composition (the chain rule) to symmetric monoidality under the smash product. The authors note a recent announced proof by Blom, but present it as a conjecture in this text to highlight its significance for comparing operadic structures and right-module presentations of derivatives.