The CP violation observed in the hadronic decays of charmed mesons remains a puzzling open question for theorists. Calculations relying on the assumption of inelastic final-state interactions occurring between the pairs of pions and kaons fall short of the experimental value. It has been pointed out that a third channel of four pions can leave imprints on the CP asymmetries of the two-body decays. At the same time, plenty of data are available for the $4\pi$ decays of charmed mesons, as well as for the rare decays $D^0\to\pi^+\pi^-\ell^+\ell^-$. With this motivation, we study the cascade topology $D^0\to a_1(1260)^+(\to \rho(770)^0\pi^+)\,\pi^-$
, which has been measured to contribute significantly to the $4\pi$ decays, and estimate its effect on the branching ratio of the rare decays. We also explore the possibility of this topology contributing to the decay amplitude of $D^0\to\pi^+\pi^-$ and by extension to the related CP asymmetry.