The high-temperature quantum-classical asymptotics conjecture

Let $Z_q(\beta,h)$ and $Z_c(\beta)$ be the quantum and classical statistical sums, and let $E_q(\beta,h)$ and $E_c(\beta)$ be the corresponding quantum and classical mean energies. Here $h>0$ is fixed and $\beta$ is the inverse temperature. High-temperature quantum-classical asymptotics conjecture. The following asymptotic equalities hold as $\beta\to+0$:

$$(2\pi h)^N Z_q(\beta,h)\sim Z_c(\beta),$$ $$E_q(\beta,h)\sim E_c(\beta).$$

These assertions propose the classical high-temperature limit for both the partition function and mean energy, but the source supplies no resolution.