The Markovian-family conjecture for pseudo-Anosov flows

Let $\Phi$ be a pseudo-Anosov flow, let $\mathcal{P}$ be its bifoliated plane, and let $\mathcal{R}$ be a Markovian family for $\Phi$. A reduced Markov partition of $\Phi$ is a Markov partition with the reduction specified in the source, and its lifts to $\mathbb{R}^3$ form a family of rectangles whose projections to $\mathcal{P}$ can be considered. Markovian-family conjecture. Every Markovian family of $\Phi$ corresponds to the projection on $\mathcal{P}$ of the family of lifts on $\mathbb{R}^3$ of the rectangles of a reduced Markov partition of $\Phi$. Markovian families are proposed as analogues of Markov partitions inside the bifoliated plane, and the conjecture asserts that every such family arises from a reduced Markov partition.