Ampleness conjecture for log canonical varieties

Let $X$ be a projective log canonical variety with numerically trivial canonical divisor, so that $K_X\equiv 0$, and let $L$ be a strictly nef $\mathbb Q$-Cartier divisor on $X$. Ampleness conjecture. The divisor $L$ is ample. This is the $K_X\equiv 0$ special case of the log version of Serrano's conjecture with zero boundary. The paper proves the claim for non-canonical singularities in dimension three and gives partial results for canonical Calabi--Yau threefolds; the remaining cases are open.