Additivity conjecture for minimum output entropy of quantum channels

Let a quantum channel be a linear completely positive trace-preserving map $\Phi:\mathcal{M}_n(\mathbb{C})\to\mathcal{M}_k(\mathbb{C})$. Its minimum output entropy is

$$H_{\min}(\Phi)=\min_{\substack{\rho\in\mathcal{M}_n(\mathbb{C}),\ \rho\geqslant 0\\ \operatorname{Tr}\rho=1}}H(\Phi(\rho)),$$

where $H(\rho)=-\operatorname{Tr}(\rho\log\rho)$. Additivity conjecture. For all quantum channels $\Phi_1$ and $\Phi_2$, one has

$$H_{\min}(\Phi_1\otimes\Phi_2)=H_{\min}(\Phi_1)+H_{\min}(\Phi_2).$$

This conjecture was disproved by Hastings, who exhibited a counterexample for the choice $\Phi_1=\overline{\Phi_2}$; consequently, the asserted additivity does not hold for all quantum channels.