On the sharp upper bound related to the weak Muckenhoupt-Wheeden conjecture

We construct an example showing that the upper bound $[w{]}_{A_1}\log (\mathrm{e}+[w{]}_{A_1})$ for the $L^1(w)\to L^{1,\mathrm{\infty }}(w)$ norm of the Hilbert transform cannot be improved in general.