Ampleness conjecture for jet differential bundles on generic complete intersections

Let $X\subset \mathbb{P}^N$ be a generic complete intersection variety of dimension $n$ and codimension $c$. For integers $k$ and $m$ as in the jet-differential construction, let $E_{k,m}\Omega_X$ denote the corresponding jet differential bundle. Jet-differential ampleness conjecture. If $k\geqslant \lceil\frac{n}{c}\rceil$, then $E_{k,m}\Omega_X$ is ample. This is presented as a stronger conjectural positivity statement related to Debarre's conjecture on ample cotangent bundles. The surrounding discussion notes that the paper proves numerical positivity and bigness results as evidence, but does not establish this ampleness assertion.