Equivalence conjecture for Picard and analytic hyperbolicity

Let $X$ be a projective manifold, and let $b[31m\ellb[0m$ be an integer with $1\leq\ell\leq\dim(X)$. The notions of $b[31m\ellb[0m$-Picard hyperbolicity and $b[31m\ellb[0m$-analytic hyperbolicity are the corresponding intermediate hyperbolicity properties.

Picard–analytic equivalence conjecture. The manifold $X$ is $b[31m\ellb[0m$-Picard hyperbolic if and only if it is $b[31m\ellb[0m$-analytically hyperbolic.

The conjecture is proposed after Picard hyperbolicity is shown to imply the extension property needed in the paper. Its resolution would identify the two intermediate hyperbolicity frameworks for projective manifolds.