Restriction conjecture for automorphisms of large power monoids

Let $H$ be a cancellative monoid. Write $\mathcal P(H)$ for its large power monoid and $\mathcal P_1(H)$ for the submonoid consisting of the relevant one-containing subsets. An automorphism of $\mathcal P(H)$ restricts to an automorphism of $\mathcal P_1(H)$ when it maps $\mathcal P_1(H)$ bijectively onto itself.

Restriction conjecture. Every automorphism of $\mathcal P(H)$ maps $\mathcal P_1(H)$ bijectively onto itself and therefore restricts to an automorphism of $\mathcal P_1(H)$.

This is proposed as a key step toward extending the inner-automorphism phenomenon to broader classes of cancellative commutative monoids. No resolution is supplied in the source.