Proposition: A regional channel does not determine its device

There are two gauge-invariant devices supported on $\{1,2\}$ that agree on $\mathfrak{A}(\{1,2\})$ but disagree on $\mathfrak{M}$.

Proof. The kinematic unitary $U=X_1{X}_2$ is supported on $\{1,2\}$ and commutes with every Gauss operator. Its physical image is $F$. The channels

agree on $\mathfrak{A}(\{1,2\})$, because $F$ is central there. But for the cross-boundary observable $W=Z_1{Z}_3$,