The NCHO quasi-partition-function conjecture

Let $Q$ be the non-commutative harmonic oscillator (NCHO), with eigenvalues $\{\lambda_j\}_{j\geq1}$, partition function

$$Z_Q(t):=\sum_{j=1}^{\infty}e^{-\lambda_jt},$$

and quasi-partition function

$$\widetilde{Z}_Q(t):=\sum_{k=0}^{\infty}(-1)^k\frac{\zeta_Q(-k)}{k!}t^k+\frac{\alpha+\beta}{\sqrt{\alpha\beta(\alpha\beta-1)}}t^{-1}.$$

NCHO quasi-partition-function conjecture. The following are expected: (1) $\widetilde{Z}_Q(t)=Z_Q(t)$; and (2), under (1), if

$$Z_Q(t)=2\sum_{k=0}^{\infty}(-1)^k\frac{B_{Q,k}}{k!}t^{k-1},$$

then $B_{Q,k}$ has the same asymptotic behavior as the Bernoulli polynomial $B_k(\tau)$ at $\tau=\frac12$ as $k\to\infty$. The first assertion identifies the regularized quasi-partition function with the spectral partition function of the NCHO, while the second controls the Taylor expansion at the origin and the resulting meromorphic continuation of the spectral zeta function. The source presents both assertions as expected and gives no resolution.