Ambro's nonvanishing conjecture

Let $X$ be a smooth projective variety of dimension $n$ over $\a0\mathbb{C}$, and let $K_X$ be its canonical divisor. Ambro's conjecture. If $D$ is an ample divisor on $X$ and $D-K_X$ is nef and big, then $H^0(X,D)\neq 0$. This conjecture remains unproved in full generality, although it is known for surfaces and several classes of threefolds.