Conjectured beta-dependent sum rule for level-spacing autocovariances

Let $\delta I_{\ell}^{(\beta)}$ be the autocovariances of level spacings in the ${Sine}_\beta$ process, and let $C_\beta^{(1)}$ denote the corresponding weighted sum. For $\beta=1$ and $4$, let $v_\beta$ be defined by

$$v_\beta=\begin{cases}-\dfrac{\pi^2}{8},&\beta=1,\\log 2+\dfrac{\pi^2}{8},&\beta=4.\end{cases}$$

Level-spacing sum-rule conjecture.

$$C_\beta^{(1)}=\sum_{\ell=1}^{\infty}\ell\left(\delta I_\ell^{(\beta)}+\frac{1}{\beta\pi^2\ell^2}\right)=\frac{1}{12}-\frac{1}{\beta\pi^2}\bigl(v_\beta+\log(2\pi)\bigr).$$

The formula extends the proved $\beta=2$ sum rule to the orthogonal and symplectic classes. The source gives no resolution evidence.