Bubbling complex projective structures with quasi-Fuchsian holonomy

For a given quasi-Fuchsian representation rho : pi(1)(S) -> PSL2C of the fundamental group of a closed surface S of genus g >= 2, we prove that a generic branched complex projective structure on S with holonomy rho and two branch points can be obtained from some unbranched structure on S with the same holonomy by bubbling, i.e. a suitable connected sum with a copy of CP1.