The log canonical ampleness conjecture for numerically positive divisors

Let $X$ be an $n$-dimensional nonsingular projective algebraic variety over the complex numbers, and let $\Delta=\sum_{i\in I}\Delta_i$ be a reduced simple normal crossing divisor, with each $\Delta_i$ prime. The divisor $K_X+\Delta$ is the log canonical divisor of the pair $(X,\Delta)$. A $\mathbf{Q}$-Cartier $\mathbf{Q}$-divisor $L$ is numerically positive when $(L,C)>0$ for every curve $C$ on $X$. Log canonical ampleness conjecture. If $K_X+\Delta$ is numerically positive, then it is ample. This is one of the four higher-dimensional ampleness conjectures considered in the paper; the supplied source gives no resolution evidence.