Monotonicity conjecture for phase-lock area portraits

Let $\omega_0>0$, $l\in\mathbb N$, $\mu>0$, $B_0=l\omega_0$, and $A_0=2\mu\omega_0$. Let $s\in\mathbb Z_{\geqslant0}$, and suppose $(B_0,A_0)\in\partial L_{s,\pm}(\omega_0)$ for an appropriately chosen sign. Suppose that $\partial L_{s,\pm}(\omega_0)$ is tangent to $\Lambda_l(\omega_0)=\{B=l\omega_0\}$ at $(B_0,A_0)$, that its germ at this point lies to the right of $\Lambda_l(\omega_0)$, and define such a point to be a left-moving tangency if the intersection disappears for $\omega>\omega_0$. Monotonicity conjecture. There are no left-moving tangencies in the model of the Josephson effect.

The conjecture describes the proposed monotone evolution of phase-lock area portraits as $\omega$ decreases. The supplied text gives a partial result excluding generalized simple intersections, but does not provide a resolution of the conjecture itself.