Petz's conjecture on scalar-curvature monotonicity for BKM metrics

Let ${\cal D}^1_n$ be the manifold of positive definite density matrices, and let $\rho\succ\sigma$ denote the majorization order. Write $\operatorname{Scal}_{f_1}(\rho)$ for the scalar curvature of the BKM metric, where $f_1$ is the operator-monotone function associated with that metric.

Petz's conjecture. The scalar curvature of the BKM metric is a Schur-increasing function; equivalently,

$$\rho\succ\sigma\Longrightarrow \operatorname{Scal}_{f_1}(\rho)\geq \operatorname{Scal}_{f_1}(\sigma).$$

The conjecture asserts monotonicity of scalar curvature under mixing and is motivated by the interpretation of scalar curvature as statistical uncertainty. The paper presents it as an open conjecture, with the $2\times2$ case and numerical evidence known at the time.