Interpreting the results on exclusive $c\rightarrow s\mu\nu$ modes

We examine the $q^2$-binned distributions of the $D\to K\mu\nu$ decay rate and of the corresponding forward-backward asymmetry which were recently measured by BESIII, showing a mild deviation from the Standard Model predictions. We point out that the proposed solution to remedy the discrepancy by turning on a complex-valued New Physics coupling is in tension with the constraints deduced from the LHC bounds on the high-$p_T$ tail of the relevant Drell-Yan process. We then show that there are several plausible scenarios that are compatible with both the measured low-energy and high-energy constraints but the selected couplings appear to be too small to be observed in the measurements of integrated observables, except for possibly the $q^2$-binned distribution of the angular observables relevant to $D_s\to \phi (\to KK) \mu\nu$ or $\Lambda_c\to \Lambda (\to p \pi) \mu\nu$ modes.