Correspondence conjecture between horizontal Goodman surgery and veering triangulations

A veering triangulation $\Delta$ on an oriented $3$-manifold $N$ determines a transitive pseudo-Anosov flow $\phi^t(\Delta,M)$ on a closed oriented $3$-manifold $M$, and a transitive pseudo-Anosov flow $\phi^t$ on a closed oriented $3$-manifold $M$ determines a veering triangulation $\Delta(\phi^t,\mathcal C)$ after removing a collection of closed orbits $\mathcal C$. A horizontal surgery curve is a curve satisfying the relevant horizontal-surgery conditions for the corresponding flow or triangulation. Correspondence conjecture. For every veering triangulation $\Delta$ on $N$, every horizontal surgery curve $c$ for $\Delta$ is isotopic to one for $\phi^t(\Delta,M)$, and

$$\phi^t\left(\Delta_{\frac{1}{n}}(c),M_{\frac{1}{n}}(c)\right)\text{ is orbit equivalent to }\phi^t(\Delta,M)_{\frac{1}{n}}(c).$$

Conversely, for every transitive pseudo-Anosov flow $\phi^t$ and a suitable collection $\mathcal C$ of closed orbits, every horizontal surgery curve $c$ for $\phi^t$ is isotopic to one for $\Delta(\phi^t,\mathcal C)$, and

$$\Delta\left(\phi^t_{\frac{1}{n}}(c),\mathcal C\right)\text{ is isomorphic to }\Delta(\phi^t,\mathcal C)_{\frac{1}{n}}(c).$$

Here $n$ has the appropriate sign for the horizontal surgery. The conjecture would identify the dynamical and combinatorial surgery operations exactly, strengthening the correspondence between veering triangulations and pseudo-Anosov flows; the source does not state that any part of this formulation is proved.