  In [9], Migliore, Mir\'o-Roig and Nagel, proved that if $R =
\mathbb{K}[x,y,z]$, where $\mathbb{K}$ is a field of characteristic zero, and
$I=(L_1^{a_1},\dots,L_r^{a_4})$ is an ideal generated by powers of 4 general
linear forms, then the multiplication by the square $L^2$ of a general linear
form $L$ induces an homomorphism of maximal rank in any graded component of
$R/I$. More recently, Migliore and Mir\'o-Roig proved in [8] that the same is
true for any number of general linear forms, as long the powers are uniform. In
addition, they conjecture that the same holds for arbitrary powers. In this
paper we will solve this conjecture and we will prove that if
$I=(L_1^{a_1},\dots,L_r^{a_r})$ is an ideal of $R$ generated by arbitrary
powers of any set of general linear forms, then the multiplication by the
square $L^2$ of a general linear form $L$ induces an homomorphism of maximal
rank in any graded component of $R/I$.
