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

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