Small-frequency limit conjecture for the upper part of phase-lock areas

For every $r\in\mathbb Z$, let $L_r$ be the phase-lock area with rotation number $r$, let $\mathcal A_{r,1}$ be its first adjacency, and define

$$L_{r,1}=L_r\cap\{A\geq A(\mathcal A_{r,1})\}.$$

Small-frequency limit conjecture. As $\omega\to0$, the maximal distance from a point of $L_{r,1}$ to the ray $\{A\geq1\}$ on the $A$-axis tends to zero. This describes the limiting geometry of the first upper component of each phase-lock area; the source gives no resolution evidence.