  We discuss the cone theorem for quasi-log schemes and the Mori hyperbolicity.
In particular, we establish that the log canonical divisor of a Mori hyperbolic
projective normal pair is nef if it is nef when restricted to the non-lc locus.
This answers Svaldi's question completely. We also treat the uniruledness of
the degenerate locus of an extremal contraction morphism for quasi-log schemes.
Furthermore, we prove that every fiber of a relative quasi-log Fano scheme is
rationally chain connected modulo the non-qlc locus.
