Conjecture on the beta-dependent variance of ordered eigenvalues

Let $\lambda_L$ be the $L$-th ordered eigenvalue in the infinite ordered spectrum. For $\beta=1$ and $4$, define

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

Eigenvalue-variance conjecture. As $L\to\infty$,

$${\rm var}_\beta[\lambda_L]=\frac{2}{\beta\pi^2}\Big(\log(2\pi L)+v_\beta+\gamma+1\Big)-\frac{1}{6}+O\left(\frac{\log L}{L^2}\right).$$

This conjecture predicts the leading and constant terms in the variance asymptotics for the $\beta=1$ and $\beta=4$ ensembles, complementing the proved $\beta=2$ expansion. The source gives no resolution evidence.