Conjectured sharp constant for Schatten-norm commutator inequalities

Let $X$ and $Y$ be general square matrices, and let $p,q,r\geq 1$ satisfy

$$\frac{1}{p}\leq\frac{1}{q}+\frac{1}{r}.$$

Write $\lVert\cdot\rVert_s$ for the Schatten $s$-norm, and let $c_{p,q,r}$ be the smallest constant such that

$$\lVert[X,Y]\rVert_p\leq c_{p,q,r}\lVert X\rVert_q\lVert Y\rVert_r.$$

Schatten-norm commutator conjecture. In the restricted case $p=q$,

$$c_{p,q,r}=c_{p,p,r}=2^{\max(1/p,1-1/p,1-1/r)}.$$

This is a proposed sharp-constant formula extending the Böttcher–Wenzel inequality to different Schatten norms. The source presents it as a conjecture based on numerical experiments; no resolution is supplied in the given text.