The vertical amplitude conjecture for horizontal unitracks

Let $\gamma_0$ be an initial curve in the horizontal unitrack construction, and let $\gamma_n$ denote the successive curves. Define the vertical amplitude of a parametrized curve $\gamma(t)=(x(t),y(t))$, $t\in[a,b]$, by

$$V(\gamma)=\max_{t\in[a,b]}y(t)-\min_{t\in[a,b]}y(t).$$

Vertical amplitude conjecture. The sequence $V(\gamma_n)$ is unbounded. The preceding theorem establishes analogous linear bounds and unboundedness for the horizontal amplitudes $H(\gamma_n)$, whereas unbounded vertical amplitude remains open in the supplied text.