Lozenge characterization of free homotopy classes for bicontact Reeb flows

Let $X$ be an Anosov flow supported by a bicontact structure $(\xi^+,\xi^-)$. For a flow $R$, let $\mathcal{P}(R)$ denote its set of free homotopy data, and let $\mathbb{R}$-covered have its usual meaning for Anosov flows. A positive lozenge is a lozenge whose two sides at one corner are positive half-leaves; a negative lozenge is defined analogously. Lozenge characterization conjecture. There exist bitransverse Reeb flows $R^+$ and $R^-$ such that, for every $[g]\in\mathcal{P}(X)$, $[g]\in\mathcal{P}(R^+)$ if and only if $g$ fixes a positive lozenge, and $[g]\in\mathcal{P}(R^-)$ if and only if $g$ fixes a negative lozenge.

This conjecture aims to describe the relationship between free homotopy data of bicontact Reeb flows and the positive and negative lozenges in the orbit space. The source presents it as an expectation, and no general proof or disproof is supplied.