Keel–McKernan's logarithmic bend-and-break conjecture

Let $(X,D)$ be a pair consisting of a smooth projective complex variety $X$ and a reduced divisor $D$ with simple normal crossings. Let $C\subset X$ be a curve such that

$$(K_X+D)\cdotp C<0$$

and $C\nsubseteq D$.

Logarithmic bend-and-break conjecture. Through a general point of $C$ there is a rational curve meeting $D$ at most once.

This conjecture is invoked to obtain pseudoeffectiveness of $K_X+D$ in the quasi-projective setting: logarithmic bend-and-break would produce rational curves with controlled intersection with the boundary, contradicting the Poincaré growth forced by negative holomorphic sectional curvature. Its resolution status is not specified in the source.