Extension conjecture for orbit equivalences of cocompact cross sections

Let $\mathbb{R}^{d} \curvearrowright \Omega_{1}$ and $\mathbb{R}^{d} \curvearrowright \Omega_{2}$ be free Borel flows, with $d \ge 2$. Let $\mathcal{C}_{i} \subseteq \Omega_{i}$ be cocompact cross sections, and let $\zeta: \mathcal{C}_{1} \to \mathcal{C}_{2}$ be an orbit equivalence, meaning a Borel bijection such that

$$c E_{1} c' \iff \zeta(c) E_{2} \zeta(c').$$

Extension conjecture. There exists a smooth equivalence $\xi: \Omega_{1} \to \Omega_{2}$ that extends $\zeta$. This would establish the possibility of extending orbit equivalences between cross sections to smooth equivalences in the descriptive set-theoretical setting. The cited work constructs such an extension up to a compressible set, while the existence of a complete extension remains open; if always possible, it would imply smooth equivalence of all flows.