Higher-dimensional extension of the X-ray theorem for modular correspondences

Let $f\colon(S^2,P)\righttoleftarrow$ be a Thurston map, and let $F$ be its associated modular correspondence on $\mathscr M_{0,d}=\operatorname{Moduli}(S^2,P)$, as in the modular-correspondence construction. Let $\star$ be a fixed point of $F$.

Higher-dimensional extension conjecture. Theorem on X-rays holds with respect to any fixed point $\star$ of $F$.

This is a proposed extension of the paper's main theorem to modular correspondences associated with Thurston maps. The source gives no resolution of the conjecture.