The monotonicity conjecture for the quantum statistical sum

Let $Z_q(\beta,h)=\sum_{n=1}^{\infty}e^{-\beta E_n(h)}$ be the quantum statistical sum for the Schrödinger system, where $h>0$ is Planck's constant, $\beta>0$, and $N$ is the dimension of the coordinate space. Monotonicity conjecture for the quantum statistical sum. The function

$$h^N Z_q(\beta,h)$$

is monotonically decreasing with respect to $h$. This is presented as an additional conjecture beyond the proposed partition-function inequality; the source supplies no resolution. For the homogeneous potentials discussed immediately beforehand, the paper relates this monotonicity to a strict mean-energy inequality.