Ng's large-deviation conjecture for the summatory Möbius function

Let $M(x)=\sum_{n\leq x}\mu(n)$ denote the summatory Möbius function. Ng's large-deviation conjecture. There exists a constant $B>0$ such that

$$\overline{\underline{\lim}}_{x\to\infty}\frac{M(x)}{x^{1/2}(\log\log\log x)^{5/4}}=\pm B.$$

This conjecture concerns the precise scale of unusually large values of the summatory Möbius function and is associated in the paper with large-deviation analysis; it remains open.