Categorical splitting conjecture for non-degenerate braided tensor subcategories

Let $\mathcal{D}$ be a braided tensor category, and let $\mathcal{C}$ be a topologizing non-degenerated braided tensor subcategory of $\mathcal{D}$. Denote by $C_{\mathcal{D}}(\mathcal{C})$ the centralizer of $\mathcal{C}$ in $\mathcal{D}$.

Categorical splitting conjecture. There is an equivalence of braided tensor categories

$$\mathcal{D}\cong \mathcal{C}\boxtimes C_{\mathcal{D}}(\mathcal{C}).$$

This is proposed as a categorical analogue of the splitting result for factorizable Hopf algebras and would decompose $\mathcal{D}$ into the given subcategory and its centralizer. The source does not provide evidence resolving the conjecture.