  For artinian Gorenstein algebras in codimension four and higher, it is well
known that the Weak Lefschetz Property (WLP) does not need to hold. For
Gorenstein algebras in codimension three, it is still open whether all artinian
Gorenstein algebras satisfy the WLP when the socle degree and the Sperner
number are both higher than six. We here show that all artinian Gorenstein
algebras with socle degree $d$ and Sperner number at most $d+1$ satisfy the
WLP, independent of the codimension. This is a sharp bound in general since
there are examples of artinian Gorenstein algebras with socle degree $d$ and
Sperner number $d+2$ that do not satisfy the WLP for all $d\ge 3$.
