Uniqueness conjecture for semigraphical translators

Let $w$ be a width, and let $\Sigma_w$ denote the corresponding semigraphical translator: a helicoid when $w<\pi$ and a pitchfork when $w\geqslant\pi$. Uniqueness conjecture. For a given width $w$, the corresponding semigraphical translator $\Sigma_w$ is unique up to vertical translation. This conjecture concerns the uniqueness of the helicoid and pitchfork limits arising from Scherk and Scherkenoid translators; the supplied source does not indicate whether it has been resolved.