The WLP failure criterion for codimension-three equigenerated monomial almost complete intersections

Let $a\geq b\geq c$, set $d=a+b+c$, and define

$$R_{a,b,c}=\Bbbk[x,y,z]/(x^d,y^d,z^d,x^a y^b z^c).$$

The weak Lefschetz failure conjecture. The algebra $R_{a,b,c}$ fails the weak Lefschetz property if and only if both of the following conditions hold: $d=6k+3$ for an integer $k$ satisfying $a<4k+2$, and at least two of $a,b,c$ are equal.

This conjecture gives a numerical characterization of failure of the weak Lefschetz property in the codimension-three equigenerated case. It is based on computer experiments, and the equivalence remains open in the source.