Property $R_\infty$ for generalized Higman groups

We give a unified proof of property $R_\infty$ for the Higman groups $H_n$ ($n\ge 4$) and for their generalizations studied by Martin and Horbez--Huang. As a key step, we prove that the automorphism groups of these groups are acylindrically hyperbolic. As a byproduct, we obtain acylindrical hyperbolicity of the groups themselves. In addition, we give an independent proof, based on Delzant's lemma, of the criterion of Fournier-Facio and collaborators stating that if $\operatorname{Aut}(G)$ is acylindrically hyperbolic and $\operatorname{Inn}(G)$ is infinite, then $G$ has property $R_\infty$.