Tan Lei's parameter-space conjecture for the cubic Newton family

Let the $a$-family denote the parameter family of cubic polynomials used in Tan Lei's description, let $f_{{\circ \! \circ}}$ denote the specific cubic polynomial appearing in that description, and let the relevant subsets be the well-determined subset of the $a$-family and the specific subset of the filled Julia set of $f_{{\circ \! \circ}}$. Identify points by the equivalence relation generated by external rays.

Tan Lei's parameter-space conjecture. The fundamental part of the cubic Newton family is homeomorphic to the quotient of the union of those two subsets by the equivalence relation generated by external rays.

The paper gives a related post-critically finite description and proves the mating statement for a large renormalizable class, but it does not establish this full parameter-space claim; the remaining cubic Newton maps are left for further work.