We show that an Artinian quotient of an ideal
I
⊆
K
[
x
,
y
,
z
]
I \subseteq \mathbb {K}[x,y,z]
generated by powers of linear forms has the Weak Lefschetz Property. If the syzygy bundle of
I
I
is semistable, the property follows from results of Brenner-Kaid. Our proof works without this hypothesis, which typically does not hold.