Generalized Ambro nonvanishing conjecture for 1-big divisors

Let $X$ be a smooth projective variety of dimension $n$ over $\a0\mathbb{C}$, and let $K_X$ be its canonical divisor. A nef divisor is 1-big when $L^{n-1}\cdot H>0$ for some ample divisor $H$. Generalized Ambro conjecture. If $D$ is an ample line bundle such that $D-K_X$ is nef and 1-big, then $H^0(X,D)\neq 0$. The paper provides evidence by proving the statement for non-uniruled surfaces and minimal threefolds, but it remains open in general.