Triviality of automorphisms of reduced power monoids of proper numerical monoids

A numerical monoid is a submonoid $S$ of $(\mathbb N,+)$ such that $\mathbb N\smallsetminus S$ is finite. Let the reduced power monoid of $S$ be the reduced power monoid associated with the additive monoid $(S,+)$. Triviality conjecture. If $S$ is a numerical monoid properly contained in $\mathbb N$, then the automorphism group of the reduced power monoid of $S$ is trivial; equivalently, its only automorphism is the identity. This would extend the result that the reduced power monoid of $(\mathbb N,+)$ has exactly two automorphisms, namely the identity and the reversion map. The statement is presented as evidence that the automorphism result may extend to proper numerical monoids, but no resolution is given here.