The WLP characterization conjecture for level Artinian almost complete intersections

Let $R=K[x,y,z]$, and let

$$I=(x^{\alpha+t},y^{\beta+t},z^{\gamma+t},x^\alpha y^\beta z^\gamma)$$

be a level Artinian almost complete intersection, where $t>0$ and $0\leq\alpha\leq\beta\leq\gamma$. Assume that $K$ is algebraically closed of characteristic zero. The WLP characterization conjecture. The quotient $R/I$ has the WLP whenever at least one of the following holds: (a) $\alpha=0$; (b) $\alpha+\beta+\gamma$ is not divisible by $3$; (c) $\gamma>2(\alpha+\beta)$; or (d) $t<\frac{1}{3}(\alpha+\beta+\gamma)$. If $1\leq\alpha\leq\beta\leq\gamma\leq2(\alpha+\beta)$, $\alpha+\beta+\gamma\equiv0\pmod 3$, and $t\geq\frac{1}{3}(\alpha+\beta+\gamma)$, then $R/I$ fails to have the WLP if and only if $t$ is even and either (i) $\alpha$ is even, $\alpha=\beta$, and $\gamma-\alpha\equiv3\pmod 6$, or (ii) $\alpha$ is odd and either $\alpha=\beta$ with $\gamma-\alpha\equiv0\pmod 6$, or $\beta=\gamma$ with $\gamma-\alpha\equiv0\pmod 3$. In all these cases, the Hilbert function has twin peaks. The claim gives a proposed complete characterization of the weak Lefschetz property in this family, extending the preceding semistability and congruence criteria; its status is not resolved in the supplied source context.