Inner automorphism conjecture for the large power monoids of the integers

Let $\mathcal P(\mathbb Z)$ and $\mathcal P_0(\mathbb Z)$ denote the large power monoid and reduced large power monoid, respectively, of the additive group $\mathbb Z$. An automorphism of either monoid is inner if it is induced by an automorphism of $\mathbb Z$ through the augmentation map.

Inner automorphism conjecture. The automorphisms of $\mathcal P(\mathbb Z)$ are all inner, and the automorphisms of $\mathcal P_0(\mathbb Z)$ are all inner.

If true, both automorphism groups would be cyclic of order two, because the only nontrivial automorphism of $\mathbb Z$ is $x\mapsto -x$. The conjecture is presented as an open problem, with evidence suggesting that it may hold more generally for broad classes of cancellative commutative monoids.