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We compute the S p → S p norm of a general Gaussian gauge-covariant multi-mode channel for any 1 ≤ p < ∞, where S p is a Schatten space. As a consequence, we verify the Gaussian optimizer conjecture and the multiplicativity conjecture in these cases. Published by AIP Publishing. [http://dx.doi:10.1063/1.4985153 fatcat:6bqamqfpena3bn6frjhw62a23a