The quasiconformal formulation of combinatorial rigidity

Let ${\cal F}$ be the exponential family or a family of unicritical polynomials. Two maps in ${\cal F}$ are combinatorially equivalent when they have the same combinatorial data, such as orbit portraits. Combinatorial rigidity conjecture, quasiconformal version. Any two combinatorially equivalent functions $f_c,f_{c'}\in{\cal F}$ are quasiconformally conjugate. This is presented as another version of the combinatorial rigidity conjecture, originally stated in the cited literature. The survey does not give a resolution for the exponential and unicritical families in this formulation.