  In [HSS], Conjecture 5.5.2, Harbourne, Schenck and Seceleanu conjectured
that, for $r=6$ and all $r\ge 8$, the artinian ideal $I=(\ell _1^2,\dots
,l_{r+1}^2)\subset K[x_1, \dots ,x_r]$ generated by the square of $r+1$ general
linear forms $\ell _{i}$ fails the Weak Lefschetz property. This paper is
entirely devoted to prove this Conjecture. It is worthwhile to point out that
half of the Conjecture - namely, the case when the number of variables $r$ is
even - was already proved in [mmn], Theorem 6.1.
