The isomorphism conjecture for power monoids of numerical monoids

Let $S$ and $S'$ be numerical monoids. For a numerical monoid $T$, let $\mathcal P_{\mathrm{fin},0}(T)$ denote the power monoid of finite subsets of $T$ containing $0$. Isomorphism conjecture. If

$$\mathcal P_{\mathrm{fin},0}(S)\cong\mathcal P_{\mathrm{fin},0}(S'),$$

then $S=S'$. The corresponding assertion for $\mathcal P_{\mathrm{fin}}(S)$ is known by the theorem cited immediately before this conjecture; the version for power monoids of finite subsets containing $0$ is presented as an unresolved continuation.