The interval conjecture for consecutive constrictions of phase-lock areas

For $r\in\mathbb Z$, let $L_r$ be the phase-lock area of rotation number $r$. In its upper part, enumerate the constrictions $\mathcal A_{r,1},\mathcal A_{r,2},\ldots$ by increasing ordinate, and let $[\mathcal A_{r,j},\mathcal A_{r,j+1}]$ denote the interval between consecutive constrictions.

The interval conjecture. For every $r\in\mathbb Z$ and $j\in\mathbb N$,

$$[\mathcal A_{r,j},\mathcal A_{r,j+1}]\subset L_r.$$

The claim describes the expected connected geometry between consecutive constrictions. The source presents it as an open conjecture and discusses the paper's results as partial progress toward it.