The right-of-axis conjecture for phase-lock areas

Let $\omega>0$, let $r\in\mathbb N$, and let $L_r$ be the phase-lock area of rotation number $r$. Its axis is

$$\Lambda_{r-1}=\{B=(r-1)\omega\}.$$

The right-of-axis conjecture. Each phase-lock area $L_r$ lies to the right of $\Lambda_{r-1}$, namely

$$B\big|_{L_r}>(r-1)\omega.$$

The source relates this claim to the conjecture identifying the upper phase-lock area with a vertical ray, but does not establish it independently; it is therefore open in the supplied text.