  We show that the current values of $R_K^\mathrm{exp}$ and
$R_{K^\ast}^\mathrm{exp}$ can be accommodated by allowing a nonzero New Physics
coupling $\delta C_9^{\mu\mu}$ to be complex, both in the scenario in which
only $\delta C_9^{\mu\mu}$ is affected, and in the scenario with complex
$\delta C_{9,10}^{\mu\mu}$ satisfying $\delta C_{9}^{\mu\mu}=-\delta
C_{10}^{\mu\mu}$. A presence of the weak CP-violating phase can then be tested
by measuring the CP-asymmetry, $\mathcal{A}_\mathrm{CP}$. We show that this
asymmetry is enhanced around the peak of each $c\bar c$-resonance, and in fact
more pronounced in the close vicinity of $J/\psi$ and $\psi(2S)$. Therefore,
measuring $\mathcal{A}_\mathrm{CP}$ before and after the resonances' peak could
be revelatory of the CP-violation that originates from beyond the Standard
Model, or to be a significant constrain when building a realistic scenario of
New Physics.
