Monotonicity of internal counts for large multiplicity

Prove that the number of internal numerical semigroups of multiplicity m and genus k is at most the number with the same multiplicity and genus k+1 whenever k≥m−1 and m≥7.

Background

The authors note that the proposed monotonicity fails for multiplicity five in the displayed data. They therefore formulate a restricted conjecture for multiplicities at least seven.

References

If $k\geq m-1$ and $m\geq 7$, then $#\mathscr{I}(\text{mul}=m,\text{ gen}=k) \leq #\mathscr{I}(\text{mul}=m,\text{ gen}=k+1).$

— Internal numerical semigroups  (2608.19984 - Casas et al., 20 Aug 2026) in Section 6, Internal numerical semigroups with fixed multiplicity