What is the limit inferior of $p_n^2/ (\log p_n) \left\lvert 1-e^\gamma\log(p_n+\log^2p_n+\varepsilon_n)\prod_1^n (1-1/p_k)\right\rvert$?

Question

Let $p_n$ be the $n$-th prime number. The $\varepsilon_n:=\varepsilon(p_n)$ in the title is an infinitesimal sequence chosen so that, replacing $p_n$ with $x$, we have$$\lim_{x\to+\infty} \left(\left(1-\frac1x\right)\frac{\log(x+\log^2x+\varepsilon(x))}{\log(x+\log^2x-\log x)}\right)^{x^2}=+\infty.$$Among the possibilities I particularly liked $\dfrac{\log\log\log p_n}{p_n}\log^3p_n;$ so playing with Mertens' product formula I got interested in the asymptotic behaviour of$$s_n=\frac{p_n^2}{\log p_n}\left\lvert1-e^\gamma\log\left(p_n+\log^2p_n+\frac{\log\log\log p_n}{p_n}\log^3p_n\right)\prod_{k=1}^n\left(1-\frac1{p_k}\right)\right\rvert.$$ Is $\liminf s_n$ known?