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

Let ${\cal P}^1_n$ be the manifold of positive probability vectors. 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 $\alpha$-geometry at $\rho\in{\cal P}^1_n$. Let $\succ$ denote majorization and “Schur-increasing” and “Schur-decreasing” refer to this order.

Commutative alpha-geometry conjecture. Suppose $n>2$. 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.

The claim extends the explicitly computable flat case $p=1$ and spherical case $p=2$ to the remaining commutative alpha-geometries. The paper presents it as motivated by the geometry of unit $L^p$ spheres; no resolution is supplied.