Proposition: Reference-stable unconditioned no-signalling

For a fixed state reaching the two wing interfaces, replacement of the left complete local instrument by any other trace-preserving left local instrument does not change the unconditioned right output, including any untouched reference. The analogous right-to-left statement holds.

Proof. Let ${\Lambda }_L$ be the complete left channel, including its record. Its adjoint is unital. For any observable $B$ on the right output and untouched reference,

Hence every such expectation is independent of ${\Lambda }_L$. Equality of expectations for all output observables proves equality of the right reduced state.