LeBrun–Mason type correspondence conjecture for space-like Zoll Einstein–Weyl structures

Let $X$ be a real manifold equipped with an indefinite Weyl structure $([g],\nabla)$, meaning a conformal class $[g]$ of indefinite metrics and a connection satisfying $\nabla g=a\otimes g$ for some $1$-form $a$. It is Einstein–Weyl if the symmetrized Ricci tensor satisfies $R_{(ij)}=\Lambda g_{ij}$, and it is space-like Zoll if every maximal space-like geodesic is closed. A totally real embedding is an embedding of a real manifold whose tangent spaces contain no complex lines.

LeBrun–Mason type correspondence conjecture. There is a natural one-to-one correspondence, at least in a neighborhood of the standard objects, between equivalence classes of space-like Zoll Einstein–Weyl structures on $S^2\times\mathbb R$ and equivalence classes of totally real embeddings

$$\iota:\mathbb{C}\mathbb{P}^1\hookrightarrow\mathbb{C}\mathbb{P}^1\times\mathbb{C}\mathbb{P}^1.$$

This conjectures a local twistor correspondence between Einstein–Weyl geometries satisfying the global space-like Zoll condition and totally real submanifolds of the relevant complex twistor space. The qualification “in a neighborhood of the standard objects” reflects that the proposed correspondence is established only perturbatively near the model case.