Quantum spectral conjecture for finite alternating Hilbert operators

For a positive integer $N$, define the quantum alternating Hilbert matrix

$$A_N^{\mathrm{quant}}=\left(\frac{\sin\frac{\pi}{N}}{\sin\frac{\pi}{N}(m-n)}\right)_{m,n=1,\ldots,N}.$$

Its spectrum is conjectured to be

$$\operatorname{Spect}(A_N^{\mathrm{quant}})=\left\{\begin{array}{ll}\left\{\pm i\,2\left(\sin\frac{\pi}{N}\right)\left(k-\frac12\right)\mid k=1,\dots,\frac{N}{2}\right\}&\text{for }N\text{ even},\\[4pt]\{0\}\cup\left\{\pm i\,2\left(\sin\frac{\pi}{N}\right)k\mid k=1,\dots,\frac{N-1}{2}\right\}&\text{for }N\text{ odd.}\end{array}\right.$$

Quantum spectral conjecture. The displayed formula gives the complete spectrum of the quantum alternating Hilbert matrix, with separate forms according to the parity of $N$. The source describes this as a quantum analogue of the preceding asymptotic periodicity conjecture and notes that it might imply that conjecture; no resolution is supplied.