Dual Goodwillie calculus conjecture for simplicial model categories

Let $\mathcal{C}$ be a simplicial model category in which Goodwillie calculus can be performed, with stabilization functors $\Sigma^\infty_\mathcal{C}$ and $\Omega^\infty_\mathcal{C}$. Write $\partial_*(\Sigma^\infty_\mathcal{C}\Omega^\infty_\mathcal{C})$ for the prospective cooperad of generalized Goodwillie derivatives of $\Sigma^\infty_\mathcal{C}\Omega^\infty_\mathcal{C}$, and write $\partial_*I_\mathcal{C}$ for the corresponding derivatives of the identity functor.

Dual Goodwillie calculus conjecture. There should be a cooperad $\partial_*(\Sigma^\infty_\mathcal{C}\Omega^\infty_\mathcal{C})$ and an equivalence of operads

$$\partial_*I_\mathcal{C} \simeq C\bigl(\partial_*(\Sigma^\infty_\mathcal{C}\Omega^\infty_\mathcal{C})\bigr).$$

The conjecture is imprecisely stated because a proper formulation requires coloured operads of spectra, which are outside the scope of the paper; the authors expect their results to extend to that setting.