Acylindrical hyperbolicity conjecture for automorphism groups of virtually cocompact special groups

Let $G$ be a virtually cocompact special group. An acylindrically hyperbolic automorphism-group conjecture. If $G$ is acylindrically hyperbolic, then its automorphism group

$$\operatorname{Aut}(G)$$

is acylindrically hyperbolic. The paper proves the analogous statement for graph products, and suggests that the same arguments may extend to virtually cocompact special groups; the assertion remains unresolved here.