Tringali and Wen's automorphism conjecture for the reduced finitary power monoid of the integers

Let the reduced finitary power monoid of the additive group $(\mathbb{Z},+)$ be the quotient of the finitary power monoid by translation equivalence, and let $X\mapsto -X$ denote the map sending each finite subset $X\subseteq\mathbb{Z}$ to its additive inverse. Tringali and Wen's conjecture. The only non-trivial automorphism of the reduced finitary power monoid of $(\mathbb{Z},+)$ is given by

$$X\mapsto -X.$$

This conjecture concerns whether the displayed involution is the sole non-identity automorphism after quotienting by translations. The source records it as a conjecture left at the end of earlier work; no resolution is given here.