Charm Physics is highly topical in the current flavour landscape, especially after the announcement by LHCb of the measurement of direct CP asymmetries in the separate decays of $D^0\rightarrow K^+K^-$ and consequently $D^0\rightarrow \pi^+\pi^-$, preceded by the measurement of the difference of these two asymmetries. The experimental result is extremely difficult to interpret, as the fully hadronic decays of charm entail significant QCD uncertainties, precluding tests of the Kobayashi-Maskawa mechanism in the up-type sector. In this work we address the problem of the theoretical determination of the strong amplitudes involved by considering very general properties of amplitudes, namely unitarity and analyticity. We implement these properties in two-channel dispersion relations which describe the final state interactions between the pion and kaon pairs, using data-driven parameterizations of all the strong rescattering quantities. While reproducing the experimental branching fractions of the aforementioned decays as well as of the ones related through isospin, we predict CP asymmetries which are way below the experimental values. Moreover, we argue that even without considering the specifics of the uncertainty-plagued inelasticity between the pions and kaons, the two-channel hypothesis always yields a severe underestimation of the CP asymmetries.