Sphere branched coverings and the growth rate inequality

We show that the growth inequality rate $$\mathrm{lim\hspace{0.17em}sup}\frac{1}{n}\log (\mathrm{\#}Fix(f^n))\ge \log d$$ holds for branched coverings of degree $d$ of the sphere $S^2$ having a completely invariant simply connected region $R$ with locally connected boundary, except in some degenerate cases with known couterexamples.