The weak Lefschetz property conjecture for level monomial ideals

Let $R$ be the polynomial ring in the variables used to define the ideal, over a field $K$ of characteristic zero. Let $0<\alpha\leq\beta\leq\gamma\leq2(\alpha+\beta)$, let $t\geq\frac{1}{3}(\alpha+\beta+\gamma)$, and assume that $\alpha+\beta+\gamma$ is divisible by three. Exclude $(\alpha,\beta,\gamma,t)=(2,9,13,9)$ and $(3,7,14,9)$. The weak Lefschetz property conjecture. The algebra

$$R/I_{\alpha+t,\beta+t,\gamma+t,\alpha,\beta,\gamma}$$

fails to have the weak Lefschetz property if and only if $t$ is even, $\alpha+\beta+\gamma$ is odd, and $\alpha=\beta$ or $\beta=\gamma$. The conjecture gives an explicit characteristic-zero criterion for failure of the weak Lefschetz property in this family; the two exceptional parameter tuples are excluded, and no resolution is supplied here.