Moreno-Frías–Rosales conjecture for numerical semigroup semimodules

For a numerical semigroup $S$, let $g(S)$ denote its genus and, for each $r\in\mathbb{N}_0$, let

$$\mathcal{J}(S,r):=\{\widetilde{\Delta}\mid \Delta\text{ is an }S\text{-semimodule and }g(\widetilde{\Delta})-g(S)=r\},$$

where $\widetilde{\Delta}=\Delta\cup\{0\}$ is the numerical semigroup associated with $\Delta$. Moreno-Frías–Rosales conjecture. There exists a positive integer $n_S$ such that, whenever $0\leq a<b\leq n_S$, one has

$$\#\mathcal{J}(S,a)\leq\#\mathcal{J}(S,b),$$

and, for every positive integer $n\geq n_S$,

$$\#\mathcal{J}(S,n)=\#\mathcal{J}(S,n_S).$$

The conjecture asserts that the cardinalities of these families are nondecreasing up to a threshold and then stabilize. Its status is not established by the supplied text.