Ambro–Ionescu–Kawamata effective adjunction conjecture

Let $(X,L)$ be a projective variety of dimension $n$ with at most terminal $\mathbb{Q}$-factorial singularities, where $L$ is a nef and big Cartier divisor; such a pair $(X,L)$ is called a quasi polarized pair. Let $t>0$. Effective adjunction conjecture. If

$$K_X+tL\in\overline{\operatorname{Eff}(X)},$$

then

$$H^0(X,K_X+tL)\neq 0.$$

This is an effective strengthening of the preceding nonvanishing proposition. The case $t=1$ is a version of the Ambro–Ionescu–Kawamata conjecture, known for $n\leq 3$; the case $t=n-1$ recovers a conjecture of Beltrametti and Sommese.