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We study the statistical monotonicity of the scalar curvature for thealpha-geometries on the simplex of probability vectors. From the resultsobtained and from numerical data we are led to some conjectures about quantumalpha-geometries and Wigner-Yanase-Dyson information. Finally we show that thislast conjecture implies the truth of the Petz conjecture about the monotonicityof the scalar curvature of the Bogoliubov-Kubo-Mori monotone metric.Comment: 20 pages, 2 .eps figures; (v2) section 2 rewritten, typos correcte

On the monotonicity of scalar curvature in classical and quantum information geometry