Evidence of $A_{CP}(D^0\to \pi^+\pi^-)$ implies observable $CP$ violation in the $D^0\to \pi^0\pi^0$ decay

Inspired by the recent measurement of $CP$ asymmetry in the individual mode on LHCb, we study $CP$ asymmetry of the $D\to \pi\pi$ system in the isospin and topological analysis. The ratio between penguin and tree amplitudes $P/(T+C)$ in the $D\to \pi\pi$ system is found to be greater than two in most values of the relative strong phase. And $D^0\to \pi^0\pi^0$ is a potential mode to reveal the $CP$ asymmetry of the order of $10^{-3}$, which would be observed by Belle II in the future. The large $CP$ asymmetry in the $D\to \pi\pi$ system might be understood in the $t$-channel final-state interaction.