Before revision:  We show how such a stabbing set can be computed for n convex polygons of constant size in the plane  in O((p-q+1)n^{4/3}\log^{ 2  + \epsilon (n ) + p^2\log(p) ) expected time.
After revision:  We show how such a stabbing set can be computed for n convex polygons of constant size in the plane in O((p-q+1)n^{4/3}\log^{ 2 n(\log\log n)^{1/3 + \epsilon (n ) + p^2\log(p) ) expected time.