Eventual fixed-perimeter Franklin inequality

Prove that for every integer j0 and every integer k, the fixed-perimeter partition count FDj,k(n) is at least FOj,k(n) for all sufficiently large n.

Background

The paper defines FOj,k(n) as the number of fixed-perimeter partitions of perimeter n having exactly j parts divisible by k, and FDj,k(n) as the number having exactly j part sizes whose multiplicity is at least k. Theorem 1.1 proves equality FOj,2(n)=FDj,2(n) for k=2, extending earlier fixed-perimeter identities.

For k, the two counting functions are not equal for all n. The authors report computational evidence suggesting that FDj,k(n) eventually dominates FOj,k(n), but they do not prove this asymptotic inequality.