Luo–Rao–Xiong conjecture on local connectedness of components under OSC without rotations/reflections

Prove that for any planar self-similar iterated function system satisfying the open set condition and involving neither rotations nor reflections, every connected component of its attractor is locally connected.

Background

The paper investigates the local connectedness of components of self-similar sets in the plane. Luo, Rao, and Xiong previously conjectured that for planar self-similar iterated function systems (IFS) satisfying the open set condition and using no rotations or reflections, each connected component of the attractor is locally connected.

This work constructs a homogeneous Lalley–Gatzouras-type counterexample satisfying the open set condition and involving no rotations or reflections, where a particular component of the attractor is not locally connected, thereby disproving the conjecture.