Orbifold rational connectedness and rational curves conjecture

Let $(X,D)$ be a smooth projective orbifold pair. A $D$-rational curve is an orbifold morphism from a curve $(C,D_C)$ to $(X,D)$ with $\deg(K_C+D_C)<0$; the pair is uniruled (respectively, rationally connected) if an irreducible orbifold rational curve passes through a generic point (respectively, generic pair of points), and it is weakly uniruled if a rational curve $C$ through a generic point satisfies $(K_X+D)\mathbin{.}C<0$. The pair is slope rationally connected if it satisfies the slope-rational-connectedness condition defined in the paper.

Orbifold rational curves conjecture. 1. $(X,D)$ is uniruled if and only if $K_X+D$ is not pseudo-effective. 1'. $(X,D)$ is uniruled if and only if it is weakly uniruled. 2. $(X,D)$ is rationally connected if and only if it is slope rationally connected.

These conjectures seek a characterization of orbifold uniruledness and rational connectedness through the positivity of the orbifold canonical divisor and slope geometry. The source notes that the conjectures are related to earlier conjectures concerning orbifold rational curves and that only a weak partial solution is obtained under an additional covering-by-negative-curves question.