Virtual retract conjecture for iterated monodromy groups of post-critically finite polynomials

Let $f\in \mathbb{C}[x]$ be a post-critically finite rational function. Write $\mathrm{IMG}(f)$ for its iterated monodromy group, and say that it has $(\mathrm{LR})$ when it has the property denoted by $(\mathrm{LR})$ in the source. The Thurston orbifold of $f$ is euclidean when its Euler characteristic satisfies $\chi(\nu_f)=0$. Virtual retract conjecture. The following are equivalent:

$$\mathrm{IMG}(f)\text{ has }(\mathrm{LR});$$ $$\mathrm{IMG}(f)\text{ is virtually abelian};$$ $$f\text{ has a euclidean orbifold}.$$

This conjecture proposes that the virtual-retract property, virtual abelianness, and euclidean Thurston-orbifold geometry coincide for these iterated monodromy groups. The surrounding discussion motivates it by contrasting euclidean maps with hyperbolic rational maps, whose iterated monodromy groups are expected to have weakly branch closures.