Fan–Tringali's sets-of-lengths conjecture for power monoids of numerical monoids

Let $S$ be a numerical monoid, and let $\mathcal P_{\mathrm{fin}}(S)$ denote its power monoid of finite nonempty subsets. For $A\in\mathcal P_{\mathrm{fin}}(S)$, let $\mathsf L(A)$ be the set of factorization lengths of $A$ in this monoid. Fan–Tringali's conjecture. For every numerical monoid $S$ and every finite nonempty subset $L\subset\mathbb N_{\ge 2}$, there is a finite nonempty set $A\subset S$ such that

$$\mathsf L(A)=L.$$

This conjecture asserts that power monoids of numerical monoids realize every finite nonempty subset of $\mathbb N_{\ge 2}$ as a set of lengths, extending the phenomenon known for several broad classes of monoids. Its status is not resolved in the supplied source.