The decay Formula omitted is studied in a sample of quantum-correlated Formula omitted pairs, based on a data set corresponding to an integrated luminosity of 2.93 fb Formula omitted collected at the Formula omitted resonance by the BESIII experiment. The asymmetry between Formula omitted-odd and Formula omitted-even eigenstate decays into Formula omitted is determined to be Formula omitted, where the first uncertainty is statistical and the second is systematic. This measurement is an update of an earlier study exploiting additional tagging modes, including several decay modes involving a Formula omitted meson. The branching fractions of the Formula omitted modes are determined as input to the analysis in a manner that is independent of any strong phase uncertainty. Using the predominantly Formula omitted-even tag Formula omitted and the ensemble of Formula omitted-odd eigenstate tags, the observable Formula omitted is measured to be Formula omitted. The two asymmetries are sensitive to Formula omitted, where Formula omitted and Formula omitted are the ratio of amplitudes and phase difference, respectively, between the doubly Cabibbo-suppressed and Cabibbo-favoured decays. In addition, events containing Formula omitted tagged by Formula omitted are studied in bins of phase space of the three-body decays. This analysis has sensitivity to both Formula omitted and Formula omitted. A fit to Formula omitted, Formula omitted and the phase-space distribution of the Formula omitted tags yields Formula omitted, where external constraints are applied for Formula omitted and other relevant parameters. This is the most precise measurement of Formula omitted in quantum-correlated Formula omitted decays.