Internal semigroups dominate leaf semigroups at fixed genus

Prove that the number of leaf numerical semigroups of genus k is at most the number of internal numerical semigroups of genus k for every nonnegative integer k.

Background

The paper separates numerical semigroups at each genus into internal and leaf semigroups. This conjecture predicts that the internal class is never smaller than the leaf class.

References

If $k\in\mathbb{N}$, then $#{\mathscr{L}(\text{gen}=k)} \leq #{\mathscr{I}(\text{gen}=k)}$.

— Internal numerical semigroups  (2608.19984 - Casas et al., 20 Aug 2026) in Conjecture 8, Section 2

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

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