The alternating grid system periodicity and continued-fraction tiling conjecture

Let $R_+$ denote the positive real numbers, and let $A_s=(R_{1,0}R_{s,0})^2$ be the alternating grid system. Let $Q_s$ be the square of $G_{s,0}$ whose bottom-left vertex is the origin. Alternating grid system conjecture. For every $s\in R_+$, almost every point of $A_s$ is periodic. Inside $Q_s$ there is a sequence of periodic islands, alternately sharing edges with the two coordinate axes, which encodes the continued fraction expansion of $s$. This conjecture concerns the proposed relationship between the periodic-island structure and continued fractions; the supplied text gives motivating examples but no resolution.