The geometric realization conjecture for Markovian families of transitive Anosov flows

Let $\Phi$ be a transitive Anosov flow on a $3$-manifold $M$, let $\widetilde{\Phi}$ be its lift to $\widetilde{M}=\mathbb{R}^3$, and let $\mathcal{P}$ be the bifoliated plane associated with $\Phi$. A Markovian family is a family of rectangles in $\mathcal{P}$ satisfying the Markovian-family conditions defined in the paper. Geometric realization conjecture. Every Markovian family corresponds to the projection on $\mathcal{P}$ of the lift on $\mathbb{R}^3$ of a Markov partition of $\Phi$. This conjecture predicts that all Markovian families arise from Markov partitions of the flow, rather than being merely combinatorial structures on the bifoliated plane; whether every Markovian family has this realization remains open.