Intermediate-growth conjecture for dendrite Julia-set quadratic polynomials

Let $P$ be a postcritically-finite non-renormalizable quadratic polynomial whose Julia set $\mathcal{J}_P$ is a dendrite. Intermediate-growth conjecture. The iterated monodromy group $\operatorname{IMG}(P)$ is of intermediate growth. This conjecture formalizes the expectation arising from the example $P_1(z)=z^2+i$, whose iterated monodromy group has intermediate growth; the paper notes that the earlier phrase “similar to” $P_1$ was imprecise and had changed meaning over time.