The codimension-three equigenerated monomial almost complete intersection SLP classification conjecture

Let $R$ be a polynomial ring in three variables over a field, and let an equigenerated monomial almost complete intersection of codimension three be generated by four monomials. The strong Lefschetz property (SLP) means that multiplication by a suitable power of a general linear form has maximal rank in every degree.

SLP classification conjecture. The equigenerated monomial almost complete intersections of codimension three with the SLP are precisely those given by

$$(x_1^a,x_2^a,x_3^a,x_1^{a-1}x_2)\quad\text{with }a\geq 2,$$ $$(x_1^a,x_2^a,x_3^a,x_1^{a-2}x_2x_3)\quad\text{with }a\geq 4,$$

$(x_1^5,x_2^5,x_3^5,x_1^2x_2^2x_3)$, and $(x_1^7,x_2^7,x_3^7,x_1^3x_2^2x_3^2)$.

The conjecture proposes a complete codimension-three classification in the equigenerated case and is based on computer calculations in Macaulay2 using the MaximalRankProperties package. The first infinite family is known to have the SLP, while the second is presented in the surrounding text as believed to have the SLP; the completeness of the list remains open.