Uniqueness conjecture for almost transverse pseudo-Anosov flows

Let $\mathcal{F}$ be a taut foliation of an atoroidal 3-manifold $M$ which is not $\mathbb{R}$-covered. A foliation is almost transverse to a flow when it satisfies the almost-transversality condition used in the paper, and a pseudo-Anosov flow has no perfect fits when its stable and unstable foliations have no perfect-fit configurations. Uniqueness conjecture. If $\mathcal{F}$ is almost transverse to a pseudo-Anosov flow $\varphi$ with no perfect fits, then $\varphi$ is the unique pseudo-Anosov flow almost transverse to $\mathcal{F}$. This generalizes the established uniqueness result for depth one foliations; the conjecture concerns the uniqueness problem beyond the $\mathbb{R}$-covered and depth one cases.