The continued-fraction length conjecture for staircase classes

Let ${\bf E}_{\delta}$ be a class with center $p/q$, and let $\ell_{{\mathcal C}{\mathcal S}}({\bf E}_{\delta})$ denote its ${\mathcal C}{\mathcal S}$-length. Let $CF$-length denote the length of a continued-fraction expansion, and consider the two staircases associated to ${\bf E}_{\delta}$.

Continued-fraction length conjecture. (i) The continued-fraction expansion of the center $p/q$ of ${\bf E}_{\delta}$ has $CF$-length $\ell_{{\mathcal C}{\mathcal S}}({\bf E}_{\delta})$. (ii) The steps of the two staircases associated to ${\bf E}_{\delta}$ have periodic continued fractions with periodic part of length $2\ell_{{\mathcal C}{\mathcal S}}({\bf E}_{\delta})$. Moreover, these periodic parts have reverse cyclic order.

The conjecture predicts a precise correspondence between recursively defined combinatorial lengths of staircase classes and the arithmetic structure of their centers and steps. No resolution is given in the supplied text.