The noncommutative alpha-geometry conjecture on scalar-curvature Schur monotonicity

Let ${\cal D}^1_n$ be the manifold of positive definite density matrices. For $p\in\mathbb R\setminus\{0\}$, define

$$A_p(\rho)=p\rho^{1/p},$$

and set $A_\infty(\rho)=\log(\rho)$. Let $p=\frac{2}{1-\alpha}$, and denote by $\operatorname{Scal}_p(\rho)$ the scalar curvature of the induced noncommutative $\alpha$-geometry at $\rho\in{\cal D}^1_n$. Let $\succ$ denote the matrix majorization order.

Noncommutative alpha-geometry conjecture. Suppose $n\geq2$. If $p\in(1,2)$, then $\operatorname{Scal}_p$ is strictly Schur-decreasing; if $p\in(2,+\infty]$, then $\operatorname{Scal}_p$ is strictly Schur-increasing.

This is the noncommutative analogue of the commutative conjecture. The cases $p=1$ and $p=2$ are known explicitly, with constant scalar curvature, but the asserted monotonicity in the other parameter ranges remains open in the paper.