  In [MMR], Micha\l{}ek--Mir\'o-Roig give a beautiful geometric
characterization of Artinian quotients by ideals generated by quadratic or
cubic monomials, such that the multiplication map by a general linear form
fails to be injective in the first nontrivial degree. Their work was motivated
by conjectures of Ilardi and Mezzetti-Mir\'o-Roig-Ottaviani, connecting the
failure to Laplace equations and classical results of Togliatti on osculating
planes. We study quotients by quadratic monomial ideals, explaining failure of
the Weak Lefschetz Property for some cases not covered by [MMR].
