Infinite-length and nondifferentiability conjecture for blocked mixed labyrinth fractals

Let $\{{\cal A}_k\}_{k=1}^{\infty}$ be a sequence of labyrinth patterns that are both horizontally and vertically blocked, with $m_k\ge 4$, and let $L_{\infty}$ be the resulting limit set. For any two points in $L_{\infty}$, let $a\subset L_{\infty}$ be the arc connecting them. Infinite-length and nondifferentiability conjecture. The length of $a$ is infinite, and the set of points at which no tangent to $a$ exists is dense in $a$. This predicts highly irregular arcs in mixed labyrinth fractals; the supplied text gives no resolution or supporting result for the assertion.