This is a question related to an earlier one of mine, which I may answer myself eventually, as I have learnt more about the topic.
Despite what one can read on the MathWorld page about Mertens' third theorem, in 2009 Diamond and Pintz showed$$\underset{p\ \text{prime}}{\prod_{p\le x}}\left(1-\frac1p\right)^{-1}-e^\gamma\log x=\Omega_{\pm}\left(\frac{\log\log\log x}{\sqrt{x}}\right);$$on the other hand, by a theorem of Dusart, for all $x\ge2973$ $$\left\lvert\prod_{p\le x}\left(1-\frac1p\right)^{-1}-e^\gamma\log x\right\rvert <\frac{0.2}{\log x}.$$ In the end of 2014, using a probabilistic argument, Lamzouri conjectured
$$\left\lvert\prod_{p\le x}\left(1-\frac1p\right)^{-1}-e^\gamma\log x\right\rvert \sim \frac{e^\gamma}{2\pi}\frac{(\log\log\log x)^2}{\sqrt{x}}.$$ Has it been established that$$-\infty<\liminf_{x\to\infty}\frac{\sqrt{x}}{(\log\log\log x)^2}\left(\prod_{p\le x}\left(1-\frac1p\right)^{-1}-e^\gamma\log x\right)<0 \ ?$$I googled quite a lot, but I could find nothing. Has there been no progress yet?