The maximum-deviation limit conjecture for the Sine-beta counting process

Let $M_{\lambda,t}$ be the martingale process associated with the Sine$_\beta$ counting process, and let $M_{\lambda,\infty}$ denote its terminal value. For $x\geq 1$, consider the maximum over spectral parameters $0\leq \lambda\leq x$.

Maximum-deviation limit conjecture. There is a random variable $\xi$ such that

$$\max_{0\leq \lambda\leq x}(M_{\lambda,\infty})-\frac{4}{\sqrt{\beta}}\left(\log x-\frac{3}{4}\log\log x\right) \xrightarrow[x\to\infty]{\mathrm{W}} \xi.$$

This conjecture predicts a limiting distribution for the centered maximum deviation of the Sine$_\beta$ counting process. The supplied text motivates it through a Gaussian-process approximation and does not provide evidence of a resolution.