Strong Lefschetz conjecture for squares in quotients by powers of general linear forms

Let $R=k[x,y,z]$ and let $A=R/I$ be any Artinian quotient generated by powers of general linear forms. For a general linear form $L$, multiplication by $L^2$ is the graded map

$$\times L^2:A_d\longrightarrow A_{d+2}$$

for each degree $d$. Strong Lefschetz conjecture for squares. Multiplication by $L^2$ has maximal rank in all degrees. This would extend the result proved in the paper for uniform powers and beyond the almost complete-intersection case; the supplied text says that the more general assertion had not been proved there.