Bijection conjecture between almost transverse pseudo-Anosov flows and universal circles

Let $\mathcal{F}$ be a taut foliation of an atoroidal 3-manifold $M$ which is not $\mathbb{R}$-covered. Let $\mathrm{PA}(\mathcal{F})$ be the set of pseudo-Anosov flows almost transverse to $\mathcal{F}$, up to orbit equivalence, and let $\mathrm{UC}(\mathcal{F})$ be the set of $\pi_1(M)$-circle actions arising from universal circles of $\mathcal{F}$, up to semiconjugacy. For $\varphi\in\mathrm{PA}(\mathcal{F})$, write $\mathcal{O}_\varphi$ for its orbit space. Bijection conjecture. The assignment

$$\varphi\mapsto\bigl(\pi_1(M)\curvearrowright\partial\mathcal{O}_\varphi\bigr)$$

determines a bijection

$$\mathrm{PA}(\mathcal{F})\to\mathrm{UC}(\mathcal{F}).$$

This proposes a classification of almost transverse pseudo-Anosov flows by the universal-circle actions associated with the foliation. The source presents it as a strategy toward the classification problem; no resolution is supplied here.