Nonintersection conjecture for phase-lock areas and vertical lines

For $r\in\mathbb N$, let $L_{r+1}$ denote the phase-lock area with rotation number $r+1$, and define the vertical line

$$\Lambda_r=\{B=\omega r\}\subset\mathbb R^2.$$

Nonintersection conjecture. The phase-lock area $L_{r+1}$ does not intersect $\Lambda_r$. This conjecture is discussed as a possible consequence of a monodromy/Riccati statement, while the source gives no resolution evidence.