Numerical and kodaira dimensions of cotangent bundles

We conjecture the equality of the numerical and Kodaira dimensions $\nu_1^*(X)$ and $\kappa_1^*(X)$ for the cotangent bundle of compact K\"ahler manifolds $X$, generalising the classical case of the canonical bundle. We show or reduce it to the classical case of the canonical bundle for some peculiar manifolds: among them, the rationally connected ones, or resolutions of varieties with klt singularities and trivial first Chern class, in which case we show that $\nu_1^*(X)=\kappa_1^*(X)=q'(X)-dim(X)$, where $q'(X)$ is the maximal irregularity of a finite \'etale cover of $X$. The proof rests on the Beauville-Bogomolov decomposition, and a direct computation for smooth models of quotients $A/G$ of complex tori by finite groups. We conjecture that these equalities hold true, much more generally, when $X$ is `special'. The invariant $\kappa_1^*$ was already introduced and studied by Fumio Sakai in [43], the particular case of the preceding conjecture when $\kappa_1^*(X)=-dim(X)$ was introduced and studied in [29].