Campana's strengthening for pseudoeffective and uniruled varieties

Let $X$ be a smooth complex projective manifold. If $K_X$ is pseudoeffective, write $\nu_1(X):=\nu(X,\Omega_X^1)$. If $X$ is uniruled, let $r:X\to R_X$ be its maximal rationally connected fibration. The cotangent numerical-dimension strengthening.

1. If $K_X$ is pseudoeffective, then

$$\nu_1(X)\leq\kappa(X).$$

2. If $X$ is uniruled, then

$$\nu_1(X)\leq\kappa(R_X).$$

The source presents these inequalities as a strengthening implied by its preceding conjecture; they are not proved there.