Debarre's ampleness conjecture for generic complete intersections

Let $N\geqslant 2$ be an integer and let $c$ satisfy $N/2\leqslant c<N$. For positive integers $d_1,\dots,d_c$, choose generic hypersurfaces

$$H_i\subset\mathbb{P}_{\mathbb C}^N\qquad (i=1,\dots,c)$$

with $\deg H_i=d_i$, and set

$$X:=H_1\cap\cdots\cap H_c.$$

Debarre's ampleness conjecture. There is a positive lower bound $d\gg1$, depending on $N$ and $c$, such that whenever $d_1,\dots,d_c\geqslant d$, the cotangent bundle $\Omega_X$ is ample.

This conjecture extends Debarre's result for complete intersections in abelian varieties and concerns the expected positivity of cotangent bundles of sufficiently high-degree generic complete intersections in projective space. The paper presents a proof in the stated range, but the supplied parser status gives no explicit resolution evidence.