  We compute all the leading-twist azimuthal spin asymmetries in the
pion-proton induced Drell-Yan process. These spin asymmetries arise from
convolutions of the leading-twist transverse-momentum-dependent parton
distributions (TMDs) of both the incoming pion and the target proton. We employ
the holographic light-front pion wave functions for the pion TMDs, while for
the proton TMDs, we utilize a light-front quark-diquark model constructed by
the soft-wall AdS/QCD. The gluon rescattering is crucial to predict nonzero
time-reversal odd TMDs. We study the utility of a nonperturbative SU$(3)$ gluon
rescattering kernel, which extends beyond the typical assumption of
perturbative U$(1)$ gluons. Subsequently, we employ Collins-Soper scale
evolution at Next-to-Leading Logarithmic precision for the TMDs evolution. Our
predictions for the spin asymmetries are consistent with the available
experimental data from COMPASS and other phenomenological studies.
