Petz's monotonicity conjecture for the scalar curvature of the Kubo–Mori metric

Let $D_1$ and $D_2$ be states, and write $D_1\prec D_2$ when $D_1$ is more mixed than $D_2$ in the majorization order. Let $\operatorname{Scal}(D)$ denote the scalar curvature of the state space induced by the canonical correlation inner product as a Riemannian metric at the point $D$. Petz's monotonicity conjecture. If $D_1\prec D_2$, then

$$\operatorname{Scal}(D_1)>\operatorname{Scal}(D_2).$$

This conjecture predicts that scalar curvature strictly decreases as states become less mixed; its resolution is not established by the supplied text.