The LHCb collaboration has recently reported the largest CP violation effect from a single amplitude, as well as other giant CP asymmetries in several BB-meson decays into three charmless light mesons. It is also claimed that this is predominantly due to ππ→KKˉππ→KKˉ rescattering in the final state, particularly in the 1 to 1.5 GeV region. In these analyses the ππ→KKˉππ→KKˉ amplitude is by default estimated from the ππππ elastic scattering amplitude and does not describe the existing ππ→KKˉππ→KKˉ scattering data. Here we show how the recent model-independent dispersive analysis of ππ→KKˉππ→KKˉ data can be easily implemented in the LHCb formalism. This leads to a more accurate description of the asymmetry, while being consistent with the measured scattering amplitude and confirming the prominent role of hadronic final state interactions, paving the way for more elaborated analyses.