Birkar's bounded complement conjecture for log canonical Fano pairs

Let $d$ be a natural number and let $\mathfrak{R}\subset [0,1]$ be a finite set of rational numbers. For a finite set $\mathfrak{R}$, write

$$\Phi(\mathfrak{R})=\left\{1-\frac{r}{m}\mid r\in\mathfrak{R},\ m\in\mathbb{N}\right\}\cup\{1\}.$$

An $n$ complement of $K_X+B$ is a divisor $K_X+B^+$ satisfying the usual complement conditions, in particular $B^+\geq B$. Bounded complement conjecture. There exists a natural number $n$, depending only on $d$ and $\mathfrak{R}$, such that whenever $(X,B)$ is a projective log canonical pair of dimension $d$, $B\in\Phi(\mathfrak{R})$, and $-(K_X+B)$ is ample, there is an $n$ complement $K_X+B^+$ of $K_X+B$ with $B^+\geq B$. The conjecture is proved in the paper in dimensions $d\leq 3$, while it remains open in general.