Fujino's logarithmic injectivity conjecture

Let $X$ be a compact Kähler manifold and let $a$ be a simple normal crossing divisor on $X$. Let $\mathcal L$ be a semipositive line bundle on $X$, and let $s$ be a nonzero holomorphic section of $\mathcal L^{\otimes k}$ on $X$ for some positive integer $k$. Assume that $(s=0)$ contains no strata of $\Delta$.

Fujino's logarithmic injectivity conjecture. The multiplication homomorphism

$$\times s: H^i(X, \omega_X\otimes \mathcal O_X(\Delta)\otimes \mathcal L^{\otimes l}) \to H^i(X, \omega_X\otimes \mathcal O_X(\Delta)\otimes \mathcal L^{\otimes (l+k)}),$$

induced by $\otimes s$, is injective for every positive integer $l$ and every $i$.

This conjecture is presented as related to Kollár's and Enoki's injectivity theorems; the source does not state its resolution.