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