The symmetry actions and dualizability conjectures for higher categories

Let $\operatorname{Cat}$ be the category of categories, let $(\infty,n)\operatorname{Cat}$ denote an $(\infty,n)$-category of higher categories, and let $\operatorname{Ho}((\infty,n)\operatorname{Cat})$ be its homotopy category. For $n$-fold categories and dg-categories, consider the indicated enhancements of the inclusions into higher categories. Higher-categorical symmetry conjectures. The following explanations should hold: the opposite $\infty$-groupoid $\Pi_{\infty}(Y)$ gives the mirror of $Y$; the generating trivial cofibration $\ast\hookrightarrow\{\ast\leftrightarrows\ast\}$ should represent the Higgs boson or Higgs symmetry; the action of $(\mathbb{Z}/2)^n$ on $\operatorname{Ho}((\infty,n)\operatorname{Cat})$ should explain $n$-dualizable objects and the classification of TQFTs by fully dualizable objects; the analogous action for dg-categories should explain Tamarkin's proof of Deligne's conjecture and the action of $2$-discs on Hochschild cohomology; and analogous enhancements should expose symmetries for manifolds and coefficient categories in algebraic geometry. These are speculative connections to higher-categorical formulations of mirror symmetry, TQFT, Higgs phenomena, and Deligne's conjecture, and the source gives no resolution.