Extension property for maps from projective manifolds to measure-hyperbolic manifolds

Let $X$ be a projective $b[31m\dim(X)b[0m$-analytically hyperbolic manifold, let $Y$ be a projective manifold with $b[31m\dim(Y)\geq\dim(X)b[0m$, and let $H\subset Y$ be a hypersurface. A holomorphic map $f:Y\setminus H\to X$ is non-degenerate if its differential has maximal rank at some point.

Extension property. Every non-degenerate holomorphic map

$$f:Y\setminus H\longrightarrow X$$

extends to a meromorphic map $f:Y\dashrightarrow X$.

For projective manifolds of general type, a weaker extension statement is known for non-degenerate maps, while the conjecture predicts meromorphic extension under intermediate hyperbolicity. The property is an analogue of the holomorphic extension theorem for compact complex hyperbolic manifolds.