  Extreme-mass ratio inspirals (EMRIs), compact binaries with small mass ratios
$\epsilon\ll 1$, will be important sources for low-frequency gravitational wave
detectors. Almost all EMRIs will evolve through important transient orbital
$r\theta$ resonances, which will enhance or diminish their gravitational wave
flux, thereby affecting the phase evolution of the waveforms at
$O(\epsilon^{1/2})$ relative to leading order. While modeling the local
gravitational self-force (GSF) during resonances is essential for generating
accurate EMRI waveforms, so far the full GSF has not been calculated for an
$r\theta$-resonant orbit owing to computational demands of the problem. As a
first step we employ a simpler model, calculating the scalar self-force (SSF)
along $r\theta$-resonant geodesics in Kerr spacetime. We demonstrate two ways
of calculating the $r\theta$-resonant SSF (and likely GSF), with one method
leaving the radial and polar motions initially independent as if the geodesic
is nonresonant. We illustrate results by calculating the SSF along geodesics
defined by three $r\theta$-resonant ratios (1:3, 1:2, 2:3). We show how the SSF
and averaged evolution of the orbital constants vary with the initial phase at
which an EMRI enters resonance. We then use our SSF data to test a previously
proposed integrability conjecture, which argues that conservative effects
vanish at adiabatic order during resonances. We find prominent contributions
from the conservative SSF to the secular evolution of the Carter constant,
$\langle \dot{\mathcal{Q}}\rangle$, but these nonvanishing contributions are on
the order of, or less than, the estimated uncertainties of our self-force
results. The uncertainties come from residual, incomplete removal of the
singular field in the regularization process. Higher order regularization
parameters, once available, will allow definitive tests of the integrability
conjecture.
