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

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