On the qc rigidity of real polynomials

The Fatou conjecture (or the HD conjecture) asserts that any rational function can be approximated by hyperbolic rational functions of the same degree and any polynomial can be approximated by hyperbolic polynomials of the same degree. The real Fatou conjecture asserts that a real polynomial can be approximated by hyperbolic real polynomials of the same degree. A possible solution of these conjecture comes from solving the rigidity problem: any two combinatorial rational functions are quasiconformally conjugate (this state-ment is usually named the combinatorial rigidity conjecture); and a rational map other than a Latt\‘es example, carries no invariant line field on the Julia set (this is named the quasiconformal rigidity conjecture, or the NILF conjecture following Mc-Mullen and Sullivan). In [18], $\mathrm{M}\mathrm{c}\mathrm{M}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{n} $ and Sullivan reduced the Fatou conjecture to the qc rigidity problem. They stated the no invariant line field (NILF) conjecture and showed that the NILF conjectnre implies the Fatou conjecture. We don’t know whether the NILF conjecture implies the real Fatou conjecture. However, beside a solution to NILF conjecture, a solution to the combinatorial rigidity problem amon