Equality in the Miyaoka-Yau inequality and uniformization of non-positively curved klt pairs

Let $(X,\mathrm{\Delta })$ be a compact Kähler klt pair, where $K_X+\mathrm{\Delta }$ is ample or numerically trivial, and $\mathrm{\Delta }$ has standard coefficients. We show that if equality holds in the orbifold Miyaoka-Yau inequality for $(X,\mathrm{\Delta })$, then its orbifold universal cover is either the unit ball (ample case) or the affine space (numerically trivial case).