  Given an Artin group $A_\Gamma$, a common strategy in the study of $A_\Gamma$
is the reduction to parabolic subgroups whose defining graphs have small
diameter, i.e. showing that $A_\Gamma$ has a specific property if and only if
all "small" parabolic subgroups of $A_\Gamma$ have this property. Since "small"
parabolic subgroups are the puzzle pieces of $A_\Gamma$ one needs to study
their behavior, in particular their intersections. The conjecture we address
here says that the class of parabolic subgroups of $A_\Gamma$ is closed under
intersection. Under the assumption that intersections of parabolic subgroups in
complete Artin groups are parabolic, we show that the intersection of a
complete parabolic subgroup with an arbitrary parabolic subgroup is parabolic.
Further, we connect the intersection behavior of complete parabolic subgroups
of $A_\Gamma$ to fixed point properties and to automatic continuity of
$A_\Gamma$ using Bass-Serre theory and a generalization of the Deligne complex.
