Rational realization conjecture for the Thue–Morse iterated monodromy covering

Let $H_q$ be the Thue–Morse group, realized as the iterated monodromy group of the degree-$q$ branched covering described using the post-critical set $\{0,\infty,\zeta^0,\dots,\zeta^{q-2}\}$, where $\zeta=\exp(2\pi i/(q-1))$. Rational realization conjecture. The branched covering described above is isotopic to a rational map of degree $q$. This asserts that the explicitly constructed branched covering has a rational representative, strengthening its description as a topological branched covering and connecting the Thue–Morse self-similar group to complex dynamical systems. No resolution is supplied in the source.