Bigness conjecture for almost strictly nef divisors on log canonical pairs

Let $(X,\Delta)$ be a log canonical pair of dimension $n$, and let $L$ be an almost strictly nef Cartier divisor on $X$.

Bigness conjecture. The $\mathbb Q$-Cartier divisor $K_X+\Delta+tL$ is big for $t>2n$.

This conjecture is presented as closely related to the logarithmic version of Serrano's conjecture. The supplied text does not give a resolution.