Weak Malnormality Conjecture for standard parabolic subgroups of Artin groups

Let $A_\Gamma$ be an Artin group. A standard parabolic subgroup is a subgroup generated by the standard generators associated to a subgraph of the presentation graph, and a subgroup is weakly malnormal if it has finite intersection with one of its conjugates. A standard parabolic subgroup is proper when it is not all of $A_\Gamma$. Weak Malnormality Conjecture. A proper standard parabolic subgroup of $A_\Gamma$ is weakly malnormal if and only if it does not contain a standard parabolic subgroup that is a direct factor of $A_\Gamma$. In particular, if $A_\Gamma$ is irreducible, then every proper standard parabolic subgroup is weakly malnormal. This conjecture is relevant because weak malnormality of the edge subgroup in a visual amalgamated-product splitting gives an acylindrical-hyperbolicity criterion. It has been proved for several families of Artin groups, including the classes described in the paper, but remains open in general.