We show that the current values of $$R_K^\mathrm {exp}$$
R
K
exp
and $$R_{K^*}^\mathrm {exp}$$
R
K
∗
exp
can be accommodated by allowing a nonzero New Physics coupling $$\delta C_9^{\mu \mu }$$
δ
C
9
μ
μ
to be complex, both in the scenario in which only $$\delta C_9^{\mu \mu }$$
δ
C
9
μ
μ
is affected, and in the scenario with complex $$\delta C_{9,10}^{\mu \mu }$$
δ
C
9
,
10
μ
μ
satisfying $$\delta C_{9}^{\mu \mu }=-\delta C_{10}^{\mu \mu }$$
δ
C
9
μ
μ
=
-
δ
C
10
μ
μ
. A presence of the weak CP-violating phase can then be tested by measuring the CP-asymmetry, $$\mathcal {A}_\mathrm {CP}$$
A
CP
. We show that this asymmetry is enhanced around the peak of each $$c\bar{c}$$
c
c
¯
-resonance, and in fact more pronounced in the close vicinity of $$J/\psi $$
J
/
ψ
and $$\psi (2S)$$
ψ
(
2
S
)
. Therefore, measuring $$\mathcal {A}_\mathrm {CP}$$
A
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.