The $p\to q$ norm conjecture for Gaussian quantum-limited amplifiers

Let $\mathcal{A}_\kappa$ be the one-mode Gaussian quantum-limited amplifier with amplification parameter $\kappa\ge1$, let $M$ be a positive integer, and let $\hat{\omega}_z$ denote the thermal Gaussian state parametrized by $0\le z<1$. For $p\ge1$, define the Schatten norm of a positive semidefinite operator $\hat A$ by

$$\left\|\hat A\right\|_p=\left(\operatorname{Tr}\hat A^p\right)^{1/p}.$$

The $p\to q$ norm conjecture. For any $\kappa\ge1$ and any $p,q\ge1$, every quantum state $\hat\rho$ satisfies

$$\frac{\left\|\mathcal{A}_\kappa^{\otimes M}(\hat\rho)\right\|_q}{\left\|\hat\rho\right\|_p} \le \left(\sup_{0\le z<1}\frac{\left\|\mathcal{A}_\kappa(\hat\omega_z)\right\|_q}{\left\|\hat\omega_z\right\|_p}\right)^M.$$

The conjecture is known for $M=1$, for $p=q$ and arbitrary $M$, and for $p=1$ and arbitrary $M$; in the latter case the supremum is attained by the vacuum, while for $p=q$ it is asymptotically attained as $z\to1$.