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