The $K(\pi ,1)$ conjecture and acylindrical hyperbolicity for relatively extra-large Artin groups

Let $A_{\mathrm{\Gamma }}$ be an Artin group with defining graph $\mathrm{\Gamma }$. We introduce the notion of $A_{\mathrm{\Gamma }}$ being extra-large relative to a family of arbitrary parabolic subgroups. This generalizes a related notion of $A_{\mathrm{\Gamma }}$ being extra-large relative to two parabolic subgroups, one of which is always large type. Under this new condition, we show that $A_{\mathrm{\Gamma }}$ satisfies the $K(\pi ,1)$ conjecture whenever each of the distinguished subgroups do. In addition, we show that $A_{\mathrm{\Gamma }}$ is acylindrically hyperbolic under only mild conditions.