Monotonicity of total counts at fixed multiplicity

Prove that the number ns_{gm} of numerical semigroups with multiplicity m and genus g satisfies ns_{gm}≤ns_{(g+1)m} for every m≥2 and g≥m−1.

Background

The paper computes counts classified simultaneously by multiplicity and genus. The observed data motivate a monotonicity conjecture for the total, rather than only internal or leaf, counts.