Let $p_k$ denote the $k-th$ prime. Do there exist some constant $A>0$ such that for every sufficiently large $k$,

$$\sum_{p\leq p_k} \frac{1}{p} > B + \log\log p_k + \frac{A}{\log p_k}$$

where $B$ is the Mertens constant?

It is known that

$$\sum_{p\leq p_k} \frac{1}{p} = B + \log\log p_k + O\left(\frac{1}{\log p_k} \right)$$

but nothing seems to be known about the sign of the implicit constant in the Landau $O$-symbol.