Effective adjunction threshold conjecture for quasi polarized manifolds

Let $(X,L)$ be a quasi polarized manifold of dimension $n$, meaning that $X$ is a projective manifold and $L$ is a nef and big Cartier divisor. Let $s>0$. Effective adjunction threshold conjecture. One has

$$H^0(X,K_X+tL)=0$$

for every integer $t$ with $1\leq t\leq s$ if and only if $K_X+sL$ is not pseudo-effective. Equivalently, the first reduction $(X”,L”)$ is one of the pairs $(X,L)$ listed in the cited adjunction theorems. The paper presents the case $s=n-2$ as the next step after proving the corresponding result for $s=n-1$; the general assertion is left as a conjectural program.