Polydisk–Hirzebruch staircase correspondence conjecture

Define $f:\mathbb{R}\to\mathbb{R}$ by sending a number $z$ to the number whose continued fraction is obtained from the continued fraction of $z$ by subtracting one from each entry. Let $\mathrm{accum}$ and $\mathrm{accum}_H$ denote the accumulation-point functions for the polydisk and Hirzebruch-surface embedding functions, respectively. For $b$ in the domain of $\mathrm{accum}_H$, set

$$b=\mathrm{accum}_H^{-1}\circ f\circ\mathrm{accum}(\beta).$$

Staircase correspondence conjecture. For every $\beta\geq\sqrt{3}$, the function $c_\beta$ has an infinite staircase if and only if the function $c_{H_b}$ has an infinite staircase.

This conjecture proposes a correspondence between infinite staircases for polydisks and Hirzebruch surfaces under the continued-fraction transformation $f$. The source gives no proof or resolution.