Dualizability conjecture for unimodular finite braided tensor categories

Let $\mathcal{C}$ be a unimodular finite braided tensor category, and let $\operatorname{Ver}^{\zeta^{1/2}}_{p^{(n)}}$ denote a mixed higher Verlinde category. Regard these categories as objects of the Morita 4-category $\mathrm{Mor}_2(\Pr)$ of presentable braided monoidal categories. Dualizability conjecture. Every unimodular finite braided tensor category, and in particular every mixed higher Verlinde category $\operatorname{Ver}^{\zeta^{1/2}}_{p^{(n)}}$, is $(3+\frac{2}{4})$-dualizable in $\mathrm{Mor}_2(\Pr)$. Furthermore, if $p>3$ and $n=\ell=2$, the mixed higher Verlinde category $\operatorname{Ver}^{\zeta^{1/2}}_{p^{(2)}}$ is not $(3+\frac{3}{4})$-dualizable. This is motivated by the expected full extension of the associated 4-dimensional TQFTs and by the fact that dualizability of a braided tensor category is controlled by that of its symmetric center; the authors plan to return to the conjecture and related questions in future work.