Counterexample to an additivity conjecture for output purity of quantum channels

A conjecture arising naturally in the investigation of additivity of classical information capacity of quantum channels states that the maximal purity of outputs from a quantum channel, as measured by the $p$ -norm, should be multiplicative with respect to the tensor product of channels. We disprove this conjecture for $\text{p>4.79.}$ The same example (with $\mathrm{p=\infty )}$ also disproves a conjecture for the multiplicativity of the injective norm of Hilbert space tensor products.