Fujino's injectivity conjecture for log-canonical pairs

Let $X$ be a compact Kähler manifold and $D=\sum_{i=1}^{N}D_i$ be a simple-normal-crossing (snc) divisor on $X$. Let $F$ be a semi-positive line bundle on $X$, meaning that it admits a smooth Hermitian metric $h_F$ with

$$\frac{\sqrt{-1}}{2\pi}\Theta_{h_F}(F)\geq 0.$$

Let $s\in H^0(X,F^{\otimes m})$ be a section whose zero locus contains no log-canonical centers of $(X,D)$, namely no connected component of a nonempty intersection $D_{i_1}\cap\cdots\cap D_{i_k}$ of irreducible components of $D$. Fujino's injectivity conjecture. The multiplication map induced by tensoring with $s$

$$H^q(X,K_X\otimes D\otimes F)\xrightarrow{\otimes s}H^q(X,K_X\otimes D\otimes F^{\otimes(m+1)})$$

is injective for every $q$. This conjecture has been fully solved by Junyan Cao and Mihai Păun.