  The monotonicity of the scalar curvature of the state space equipped with the
Bogoliubov-Kubo-Mori metric under more mixing a state is an important
conjecture called the Petz conjecture. From the standpoint of quantum
statistical mechanics, the quantum exponential family, a special submanifold of
the state space, is central rather than the full state space. In this
contribution, we investigate the monotonicity of the scalar curvature of the
submanifold with respect to temperature for transverse-field Ising chains in
various sizes and find that the monotonicity breaks down for the chains in
finite sizes, whereas the monotonicity seems to hold if the chain is
non-interacting or infinite-size. Our results suggest that finite-size effects
can appear in the curvature through monotonicity with respect to majorization.
