Tringali–Yan conjecture on automorphisms of reduced power monoids

Recall that a numerical monoid is a submonoid $S$ of $(\mathbb{N},+)$ such that $\mathbb{N}\setminus S$ is finite. For such an $S$, let $\mathcal{P}_{0}(S)$ denote its reduced power monoid.

Tringali–Yan conjecture. If $S$ is a numerical monoid properly contained in $\mathbb{N}$, then the automorphism group of $\mathcal{P}_{0}(S)$ is the trivial group:

$$\operatorname{Aut}(\mathcal{P}_{0}(S))=\{\operatorname{id}\}.$$

This conjecture concerns whether reduced power monoids of proper numerical submonoids of $\mathbb{N}$ have any nontrivial automorphisms. The cited result establishes the analogous classification for $\mathcal{P}_{0}(\mathbb{N})$, but the status of the conjecture for every proper numerical monoid is not resolved here.