Minimum output entropy conjecture for multi-mode Gaussian quantum channels

Let $n\in\mathbb{N}$, and let $\hat{\rho}$ be a finite-entropy state of an $n$-mode Gaussian quantum system. Let $g$ be the entropy function for a single-mode thermal Gaussian state, and define

$$N(\hat{\rho})=g^{-1}\left(\frac{S(\hat{\rho})}{n}\right),$$

so that $S\left(\hat{\omega}_{N(\hat{\rho})}^{\otimes n}\right)=S(\hat{\rho})$. Denote by $\mathcal{E}_{\eta,E}$, $\mathcal{A}_{\kappa,E}$, $\tilde{\mathcal{A}}_{\kappa,E}$, and $\mathcal{N}_E$ the $n$-mode Gaussian attenuator, amplifier, phase-contravariant channel, and additive-noise channel, respectively. Minimum output entropy conjecture. Quantum Gaussian thermal input states minimize the output entropy of these channels among all input states with the same entropy; explicitly,

$$S\left(\mathcal{E}^{\otimes n}_{\eta,E}(\hat{\rho})\right)\ge n\,g\left(\eta N(\hat{\rho})+(1-\eta)E\right),$$ $$S\left(\mathcal{A}^{\otimes n}_{\kappa,E}(\hat{\rho})\right)\ge n\,g\left(\kappa N(\hat{\rho})+(\kappa-1)(E+1)\right),$$ $$S\left(\tilde{\mathcal{A}}^{\otimes n}_{\kappa,E}(\hat{\rho})\right)\ge n\,g\left((\kappa-1)(N(\hat{\rho})+1)+\kappa E\right),$$ $$S\left(\mathcal{N}^{\otimes n}_{E}(\hat{\rho})\right)\ge n\,g\left(N(\hat{\rho})+E\right).$$

The conjecture concerns the minimum output entropy at fixed input entropy for important multi-mode Gaussian channels and underlies entropy and communication-rate bounds. The supplied text does not establish its resolution or specify which parameter regimes are known, so its status remains open.