Carlen–Lieb lower rearrangement inequality conjecture for Schatten norms

Let $A,B\in M_{n\times n}(\mathbb{C})$, and let $\sigma_\downarrow(A)$ and $\sigma_\downarrow(B)$ denote their singular-value vectors arranged in decreasing order. For $1\leq p<\infty$, write $||X||_p=\operatorname{Tr}[(X^*X)^{p/2}]^{1/p}$. Carlen–Lieb's lower rearrangement conjecture. For all $1\leq p\leq 2$,

$$||A+B||_p^p+||A-B||_p^p\geq ||\sigma_\downarrow(A)+\sigma_\downarrow(B)||_p^p+||\sigma_\downarrow(A)-\sigma_\downarrow(B)||_p^p.$$

For $p>2$, the inequality reverses. This conjecture was proposed as a potential tool for proving the noncommutative analogue of Hanner's inequality. The paper states that it extends some cases but gives counterexamples to any general rearrangement inequality, so the conjecture is not supported in full generality.