Tan Lei and Shishikura's example of non-mateable degree 3 polynomials without a Levy cycle

After giving an introduction to the procedure dubbed slow polynomial mating and stating a conjecture relating this to other notions of polynomial mating, we show conformally correct pictures of the slow mating of two degree 3 post critically finite polynomials introduced by Shishikura and Tan Lei as an example of a non matable pair of polynomials without a Levy cycle. The pictures show a limit for the Julia sets, which seems to be related to the Julia set of a degree 6 rational map. We give a conjectural interpretation of this in terms of pinched spheres and show further conformal representations.