The abundance conjecture for exterior powers of the cotangent bundle

Let $X$ be a smooth compact Kähler manifold and let $\Omega_X^p$ be its sheaf of holomorphic $p$-forms. For a torsion-free coherent sheaf $\mathcal E$ on $X$, define its Kodaira and numerical dimensions by

$$\kappa(\mathcal E)=\kappa(\mathbb P(\mathcal E),\mathcal O_{\mathcal E}(1))-(\operatorname{rank}\mathcal E-1),$$ $$\nu(\mathcal E)=\nu(\mathbb P(\mathcal E),\mathcal O_{\mathcal E}(1))-(\operatorname{rank}\mathcal E-1).$$ $$\nu_p(X):=\nu(\Omega_X^p),\qquad \kappa_p(X):=\kappa(\Omega_X^p).$$

Generalized abundance conjecture. For every $X$ and every integer $p>0$,

$$\nu_p(X)=\kappa_p(X).$$

This extends the classical abundance conjecture from the canonical bundle to the cotangent sheaves. The source notes results when $\kappa_p(X)=-\infty$ and for elliptic surfaces, but gives no general resolution.