Non-tangency conjecture for phase-lock adjacencies

Let $\omega>0$, and let $(B_0,A_0)\in\mathbb R_+^2$ be an adjacency of a phase-lock area. Consider the two branches of the boundary of that phase-lock area meeting at $(B_0,A_0)$. Non-tangency conjecture. These two boundary branches cannot both be tangent at $(B_0,A_0)$ to the vertical line $\{B=B_0\}$. This conjecture is proposed in a strategy for ruling out boundary contact with certain vertical lines; the source gives no resolution evidence.