The Conjugacy Stability Problem for Parabolic Subgroups in Artin Groups

Given an Artin group $A$ and a parabolic subgroup $P$, we study if every two elements of $P$ that are conjugate in $A$, also $P$. We provide algorithm to solve this decision problem satisfies three properties conjectured be true for group. partially the has $FC$-type, totally it is isomorphic free product spherical groups. In particular, show latter case, element contained unique minimal (by inclusion) subgroup.