Local connectedness conjecture for planar self-similar sets

Let $\mathcal{F}$ be an IFS on $\mathbb{R}^2$ and let $E$ be its attractor. Suppose that $E$ satisfies the open set condition and that every contraction in $\mathcal{F}$ involves neither rotation nor reflection. Local connectedness conjecture. Every component of $E$ is locally connected (and hence path connected). The conjecture is motivated by examples in which all components are path connected, while rotations or failure of the open set condition are present in related examples; its general validity under the stated hypotheses remains open.