Divonne's higher-order jet-differential ampleness conjecture

Let $X\subset\mathbb P^n$ be the intersection of at least $n/(k+1)$ very general hypersurfaces of sufficiently high degree. Let $E_{k,m}T_X^*$ denote the bundle of invariant jet differentials of order $k$ and weighted degree $m$.

Higher-order jet-differential ampleness conjecture. The bundle map

$$E_{k,m}T_X^*\to X$$

is ample, and therefore $X$ is hyperbolic.

This is proposed as a higher-order generalization of Debarre's cotangent-bundle ampleness conjecture, since $E_{1,m}T_X^*=S^mT_X^*$. The source notes vanishing results for low jet order and explains that ampleness would imply Kobayashi hyperbolicity; the proposed generalization remains open.