Properness for circle packings and Delaunay circle patterns on complex projective structures

We consider circle packings and, more generally, Delaunay circle patterns -
arrangements of circles arising from a Delaunay decomposition of a finite set
of points - on surfaces equipped with a complex projective structure. Motivated
by a conjecture of Kojima, Mizushima and Tan, we prove that the forgetful map
sending a complex projective structure admitting a circle packing with given
nerve (resp. a Delaunay circle pattern with given nerve and intersection
angles) to the underlying complex structure is proper.