  We calculate the scalar self-force experienced by a scalar point-charge
orbiting a Kerr black hole along $r\theta$-resonant geodesics. We use the
self-force to calculate the averaged rate of change of the charge's orbital
energy $\langle\dot{E}\rangle$, angular momentum $\langle\dot{L}_z\rangle$, and
Carter constant $\langle\dot{Q}\rangle$, which together capture the
leading-order adiabatic, secular evolution of the point-charge. Away from
resonances, only the dissipative (time anti-symmetric) components of the
self-force contribute to $\langle\dot{E}\rangle$, $\langle\dot{L}_z\rangle$,
and $\langle\dot{Q}\rangle$. We demonstrate, using a new numerical code, that
during $r\theta$ resonances conservative (time symmetric) scalar perturbations
also contribute to $\langle\dot{Q}\rangle$ and, thus, help drive the adiabatic
evolution of the orbit. Furthermore, we observe that the relative impact of
these conservative contributions to $\langle\dot{Q}\rangle$ is particularly
strong for eccentric 2:3 resonances. These results provide the first conclusive
numerical evidence that conservative scalar perturbations of Kerr spacetime are
non-integrable during $r\theta$ resonances.
