Infinite-length arcs conjecture for supermixed labyrinth fractals

Let $\{\tilde{\mathcal A}_k\}_{k\ge 1}$ be a sequence of collections of labyrinth patterns. For each $k\ge 1$, every pattern in $\tilde{\mathcal A}_k$ is horizontally and vertically blocked, and let $\{m_k\}_{k\ge 1}$ be the width sequence satisfying

$$\sum_{k=1}^{\infty}\frac{1}{m_k}=\infty.$$

Infinite-length arcs conjecture. The supermixed labyrinth fractal $L_{\infty}$ generated by $\{\tilde{\mathcal A}_k\}_{k\ge 1}$ has the property that, between any two distinct points in the fractal, the arc connecting them in the fractal has infinite length.

This conjecture extends the known results for mixed labyrinth fractals, where examples with both finite and infinite connecting arcs exist. It predicts that the stated divergence condition on the widths forces every connecting arc between distinct points of a supermixed labyrinth fractal to have infinite length.