  In 2012, Migliore, the first author, and Nagel conjectured that, for all
$n\geq 4$, the artinian ideal $I=(L_0^d,\ldots,L_{2n+1}^d) \subset
R=k[x_0,\ldots,x_{2n}]$ generated by the $d$-th powers of $2n+2$ general linear
forms fails to have the weak Lefschetz property if and only if $d>1$. This
paper is entirely devoted to prove partially this conjecture. More precisely,
we prove that $R/I$ fails to have the weak Lefschetz property, provided $4\leq
n\leq 8,\ d\geq 4$ or $d=2r,\ 1\leq r\leq 8,\ 4\leq n\leq 2r(r+2)-1$.
