The converse characterization of the right-veering arc relation

Let $S$ be a surface with a set of marked points $P$, let $C$ be a relevant boundary component, and let $\alpha,\beta\in\mathcal{A}_{C}(S,P)$ be arcs. Write $\alpha\prec_{\mathrm{right}}\beta$ for the right-veering ordering, and write $\alpha\ll_{\mathrm{right}}\beta$ for the relation defined by the existence of a sequence of intermediate arcs with pairwise disjoint interiors. A boundary right $P$-bigon from $\alpha$ to $\beta$ is a boundary bigon of the type defined in the preceding discussion. Converse characterization. One has

$$\alpha\ll_{\mathrm{right}}\beta$$

if and only if

$$\alpha\prec_{\mathrm{right}}\beta$$

and there exist no boundary right $P$-bigons from $\alpha$ to $\beta$. This would give a converse to the proposition that a boundary right $P$-bigon obstructs $\alpha\ll_{\mathrm{right}}\beta$.