Proposition: Commuting channels are not an observable-locality theorem

Commuting CPTP maps do not imply that their associated observables or arbitrarily assigned region algebras commute.

Proof. On one qubit, the conjugation channels ${\mathrm{Ad}}_X$ and ${\mathrm{Ad}}_Z$ commute because $XZ=-ZX$. The observables $X,Z$ do not commute. Assigning both to separately named regions does not alter that fact. A compatible observable localization is additional data.