The doubly-presentable dualizable-object presentability conjecture

Let $\mathcal P$ be a “doubly-presentably” symmetric monoidal $(\infty,2)$-category. The $(\infty,1)$-category $\mathcal P^{\mathrm{dbl}}$ of dualizable objects and $\mathcal P$-internal left adjoints therein is defined by taking the dualizable objects and internal left adjoints in $\mathcal P$. Doubly-presentable dualizable-object presentability conjecture. The $(\infty,1)$-category $\mathcal P^{\mathrm{dbl}}$ is presentable. This is a meta-conjecture because the notion of a “doubly-presentably” symmetric monoidal $(\infty,2)$-category is not yet well-defined; the paper notes that $\mathbf{Mod}_\mathcal V(\mathbf{Pr}^{\mathrm{L}})$ and $\mathbf{Pr}^{\mathrm{L}}_\mathcal X$ are intended examples.