Modular Jordan type for $\k [x, y] / (x^m,y^n)$ for $m = 3, 4$

A sufficient condition for an Artinian complete intersection quotient $S=\k[x,y]/(x^m,y^n)$, where $\k$ is an algebraically closed field of a prime characteristic, to have the strong Lefschetz property (SLP) was proved by S. B. Glasby, C. E. Praezer, and B. Xia in \cite{GPX}. In contrast, we find a necessary and sufficient condition on $m$, $n$ satisfying $3 \le m \le n$ and $p > 2m-3$ for $S$ to fail to have the SLP. Moreover we find the Jordan types for $S$ failing to have SLP for $m \le n$ and $m = 3, 4$.