Mexican BOP get a lot of stem die back in our NM winters. You may want to see how much die back you've had based on where it is sprouting and maybe not have to cut back too far. Even in protected microclimates in Las Cruces and El Paso, they experience a far amount of die back most winters.

I live in Las Cruces and don't have any winter dieback. You might be referring to the dead flower petals (stems).

This past winter the rabbits loved the bark of my BOP. Every limb sprouted new branches and leaves. Tough plant.