Luo–Rao–Xiong's local connectedness conjecture for planar self-similar sets

Let $\Phi$ be a self-similar iterated function system on $\mathbb R^2$ and let $K$ be its attractor. The Luo–Rao–Xiong local connectedness conjecture. If $\Phi$ satisfies the open set condition and every similitude in $\Phi$ involves neither a rotation nor a reflection, then every connected component of $K$ is locally connected.

This conjecture addresses whether the connected components of planar self-similar sets must retain local connectedness under the open set condition when rotations and reflections are excluded. It was proposed after positive results for fractal squares and generalized Sierpiński carpets, while examples involving rotations show that the conclusion can fail in that broader setting.