Weak birational Serrano conjecture for almost strictly nef divisors

Let $X$ be a projective canonical variety of dimension $n$, and let $L$ be an almost strictly nef Cartier divisor on $X$, meaning that $L$ is the pullback of a strictly nef divisor under a surjective birational morphism. Let $t$ be a real number.

Weak birational Serrano conjecture. The divisor

$$K_X+tL$$

should be big for $t>2n$.

This is a birational version of the log form of Serrano's conjecture, stated in a weakened form for the paper's inductive arguments. The source presents it as an open conjecture.