Fixed-Perimeter Franklin Statistics and an Eventual Inequality

Gray, Payne, Swisher, and Watson conjectured that, for fixed integers $j \geq 0$ and $k \geq 2$, the number $FD_{j,k}(n)$ of partitions of perimeter $n$ having exactly $j$ part sizes of multiplicity at least $k$ is eventually at least the number $FO_{j,k}(n)$ having exactly $j$ distinct occurring pa