Twistor correspondence conjecture for singular self-dual Zollfrei conformal structures

Consider singular self-dual Zollfrei conformal structures on $S^2\times S^2$ with a single singular beta-surface $C$ homeomorphic to $S^2$. Let $P$ be the image of a totally real embedding $\mathbb R\mathbb P^3\rightarrow\mathbb C\mathbb P^3$ with one singular point, and consider pairs $(\mathbb C\mathbb P^3,P)$ satisfying condition $(\sharp)$.

Twistor correspondence conjecture. There is a natural one-to-one correspondence between equivalence classes of these singular self-dual Zollfrei conformal structures and equivalence classes of the pairs $(\mathbb C\mathbb P^3,P)$ satisfying condition $(\sharp)$.

This is the proposed twistor correspondence between the geometric structures and their complex-projective twistor data. The supplied text gives no resolution status.