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We forward-model mass-weighted stellar ages (MWAs) and quiescent fractions in projected phase space (PPS), using data from the Sloan Digital Sky Survey, to jointly constrain an infall quenching model for galaxies in $\log(M_{\mathrm{vir}}/\mathrm{M}_{\odot})>14$ galaxy clusters at $z\sim 0$. We find the average deviation in MWA from the MWA-$M_\star$ relation depends on position in PPS, with a maximum difference between the inner cluster and infalling interloper galaxies of $\sim 1$ Gyr. Our model employs infall information from N-body simulations and stochastic star-formation histories from the UniverseMachine model. We find total quenching times of $t_\mathrm{Q}=3.7\pm 0.4$ Gyr and $t_\mathrm{Q}=4.0\pm 0.2$ Gyr after first pericentre, for $9, 21 pages, 13 figures, submitted to MNRAS