Gonek's conjecture for the Mertens function

Let $\mu(n)$ be the Möbius function and define the Mertens function by

$$M(x)=\sum_{n\leqslant x}\mu(n).$$

Let $\overline{\varliminf}$ denote the upper limit of the lower limit. Gonek's conjecture. There exists a number $B>0$ such that

$$\overline{\varliminf}\frac{M(x)}{\sqrt{x}(\log\log\log x)^{5/4}}=\pm B.$$

The conjecture concerns the expected extreme order of the Mertens function and is presented as related to the direct Riemann hypothesis for the Riemann zeta function; the paper gives no proof or resolution.