Constrained minimum output entropy conjecture for Gaussian quantum-limited amplifiers

For $n\in\mathbb{N}$, let $\rho$ be an $n$-mode quantum state and define

$$\omega_{\rho}=\left[\omega\left(g^{-1}\left(\tfrac{1}{n}S(\rho)\right)\right)\right]^{\otimes n},$$

where $\omega_{\rho}$ is the $n$-mode thermal quantum Gaussian state with the same entropy as $\rho$. Let $\mathcal{A}_{\kappa}^{\otimes n}$ be the $n$-mode Gaussian quantum-limited amplifier with $\kappa\ge1$.

Constrained minimum output entropy conjecture. For every $n\in\mathbb{N}$, $\rho$ and $\kappa\ge1$,

$$S\left(\mathcal{A}_{\kappa}^{\otimes n}(\rho)\right)\ge S\left(\mathcal{A}_{\kappa}^{\otimes n}(\omega_{\rho})\right)=n\,g\left(\kappa g^{-1}\left(\tfrac{1}{n}S(\rho)\right)+\kappa-1\right).$$

Thermal Gaussian inputs are known to minimize the output entropy in the one-mode case, while the general $n\ge2$ conjecture remains open except for entanglement-breaking Gaussian channels.