Fixed-perimeter inequality for parts divisible by and repeated at least k times

Let $FO_{j,k}(n)$ be the number of partitions of $n$ with exactly $j$ parts divisible by $k$, and let $FD_{j,k}(n)$ be the number of partitions of $n$ with exactly $j$ parts appearing at least $k$ times. For $j\geq 0$ and $k\geq 2$, the fixed-perimeter inequality conjecture.

$$FD_{j,k}(n)\geq FO_{j,k}(n)$$

for sufficiently large $n$.

The equality $FO_{j,2}(n)=FD_{j,2}(n)$ is known for all $n$, but for $k\geq 2$ the two quantities are not equal for all $n$ in general. The stated inequality is supported by computational evidence and concerns the eventual comparison for general $k$.