Hozoori's projective deformation conjecture for Anosov flows

Let $M$ be a manifold, and consider two oriented Anosov flows on $M$. They are projectively deformation equivalent when they are related through the notion of homotopy through projectively Anosov flows described in the surrounding discussion. Hozoori's conjecture. If two oriented Anosov flows on $M$ are projectively deformation equivalent, then they are orbit equivalent. This conjecture proposes that projective deformation captures exactly the orbit-equivalence class of an Anosov flow; the source gives no resolution status.