Internal semigroups outnumber leaves at fixed Frobenius number

Prove that, for every integer Frobenius number F≥1, the number of internal numerical semigroups with Frobenius number F is greater than the number of leaf numerical semigroups with Frobenius number F.

Background

The authors enumerate internal and leaf numerical semigroups with fixed Frobenius number and use the resulting data to formulate a strict dominance conjecture.