The warming is caused by adiabatic compression as pressures rise within the weakening vortex.
What “matters” more than anything is how the PV is affected structurally. Is it a wave-1 (displacement) SSW? Or is it a wave-2 (split) SSW? And is the vortex (or daughter vortices, in the case of a split) displaced well off the pole, with high pressure taking over (hence a reversal in zonal winds)?
All of that matters more than the warming itself, which is a result of mechanical perturbation by wave activity.