Morita adjoints conjecture for higher Morita categories

Let $\mathcal{V}$ be a symmetric monoidal $(\infty,1)$-category compatible with $\mathsf{simp}^{n,\operatorname{op}}$-colimits. Let $\mathfrak{alg}_{n}(\mathcal{V})$ denote the underlying $(\infty,n)$-category of the higher Morita category of $\mathcal{V}$. Morita adjoints conjecture. The symmetric monoidal $(\infty,n)$-category $\mathfrak{alg}_{n}(\mathcal{V})$ has duals: its objects are dualizable and all $i$-morphisms have adjoints for $1\leq i<n$. In particular, all objects of $\mathfrak{alg}_{n}(\mathcal{V})$ are fully dualizable. This conjecture was proved by Gwilliam and Scheimbauer for a closely related factorization-algebra model; the claimed result for this model is expected to follow from an equivalence with that model, but that equivalence is not established here.