Carlen–Lieb upper rearrangement inequality conjecture for Schatten norms

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

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

For $p>2$, the inequality reverses. The conjecture concerns a proposed universal rearrangement bound for Schatten norms and was introduced in connection with the noncommutative Hanner inequality. The source reports counterexamples to any general rearrangement inequality, while treating several special cases and equality questions.