  We study the problem of whether an arbitrary codimension three graded
artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this
problem to checking whether it holds for all compressed Gorenstein algebras of
odd socle degree. In the first open case, namely Hilbert function
(1,3,6,6,3,1), we give a complete answer in every characteristic by translating
the problem to one of studying geometric aspects of certain morphisms from
$\mathbb P^2$ to $\mathbb P^3$, and Hesse configurations in $\mathbb P^2$.
