This Letter reports on a systematic study of β-decay half-lives of neutron-rich nuclei around doubly magic (208)Pb. The lifetimes of the 126-neutron shell isotone (204)Pt and the neighboring (200-202)Ir, (203)Pt, (204)Au are presented together with other 19 half-lives measured during the "stopped beam" campaign of the rare isotope investigations at GSI collaboration. The results constrain the main nuclear theories used in calculations of r-process nucleosynthesis. Predictions based on a statistical macroscopic description of the first-forbidden β strength reveal significant deviations for most of the nuclei with N<126. In contrast, theories including a fully microscopic treatment of allowed and first-forbidden transitions reproduce more satisfactorily the trend in the measured half-lives for the nuclei in this region, where the r-process pathway passes through during β decay back to stability.