The quasi-partition-function conjecture for quantum systems

Let $H$ be a Hamiltonian defining a quantum system, let $Z_H(t)$ be its partition function, and let $\widetilde{Z}_H(t)$ be its quasi-partition function as defined from the special values of the corresponding spectral zeta function. Quasi-partition-function conjecture. Under the assumptions in the definition of the quasi-partition function,

$$\widetilde{Z}_H(t)=Z_H(t).$$

This is proposed as a generalization of the corresponding identification for the quantum harmonic oscillator, the quantum Rabi model, and the non-commutative harmonic oscillator. The source does not provide a resolution.