The self-intersection conjecture for horizontal unitracks

Let $\gamma_0$ be an initial curve in the horizontal unitrack construction, and let $\gamma_n$ denote the successive curves. Call $\gamma_0$ trivial when it is the trivial initial curve in this construction. Self-intersection conjecture. If $\gamma_0$ is not trivial, then there exists $n$ such that $\gamma_n$ has self-intersections. The conjecture predicts that every nontrivial horizontal unitrack eventually contains a curve with a self-intersection; the supplied text states that this remains open.