Remark: References do not evade the lower bound

For an encoded state ${\rho }_{MR}$, the auxiliary systems are independent of $MR$. If the source lies outside the ancestor set, the joint destination–reference output is ${\omega }_b\otimes {\rho }_R$. For local trace-preserving payload interventions, ${\rho }_R$ is unchanged. The reference is not an extra communication wire.