The Additivity conjecture for braided and symmetric pseudomonoids

Let $\mathcal{C}$ be a Gray monoid. A braided and symmetric pseudomonoid additivity conjecture. Every braiding on $\mathcal{C}$ induces an equivalence

$$\mathrm{Alg}_{\mathsf{E}_2}(\mathcal{C})\simeq\mathrm{BrMon}(\mathcal{C})$$

of 2-categories, and every syllepsis on $\mathcal{C}$ induces an equivalence

$$\mathrm{Alg}_{\mathsf{E}_3}(\mathcal{C})\simeq\mathrm{CMon}(\mathcal{C})$$

of 2-categories.

These equivalences would provide an algebraic 2-dimensional version of the Additivity Theorem, relating higher-operadic algebra objects to braided and symmetric pseudomonoids. The source presents this characterization as future work and does not establish it here.