Maximal oscillation conjecture for the Mertens product

Let $x$ tend to infinity, and let $\gamma$ denote Euler's constant. Define the Mertens product by

$$\prod_{p\leq x}\left(1-\frac1p\right)^{-1}.$$

Maximal oscillation conjecture. As $x\to\infty$,

$$\limsup_{x\to\infty} \frac{\sqrt{x}}{(\log\log\log x)^2}\left(\prod_{p\leq x}\left(1-\frac1p\right)^{-1}-e^{\gamma}\log x\right)=\frac{e^{\gamma}}{2\pi},$$

and

$$\liminf_{x\to\infty} \frac{\sqrt{x}}{(\log\log\log x)^2}\left(\prod_{p\leq x}\left(1-\frac1p\right)^{-1}-e^{\gamma}\log x\right)=-\frac{e^{\gamma}}{2\pi}.$$

This conjecture predicts the maximal order of the oscillations of the error in the Mertens product formula, refining the known lower bound of order $\log\log\log x$ after normalization by $\sqrt{x}$. It follows the probabilistic approach used by Montgomery for the maximal size of $\pi(x)-\operatorname{Li}(x)$; the stated asymptotic extremal behavior remains unproved.