Existence of complements conjecture for log canonical pairs

Let $d$ be a positive integer, let $\Gamma\subset[0,1]$ be a DCC set, and let $(X/Z\ni z,B)$ be an $\mathbb{R}$-complementary pair of dimension $d$ whose boundary coefficients lie in $\Gamma$. An $n$-complement $(X/Z\ni z,B^+)$ means that $(X/Z\ni z,B^+)$ is log canonical,

$$nB^+\geq \lfloor(n+1)\{B\}\rfloor+n\lfloor B\rfloor,$$

and

$$n(K_X+B^+)\sim 0$$

over a neighborhood of $z$.

Existence of Complements Conjecture. There exists a positive integer $n$, depending only on $d$ and $\Gamma$, such that every such pair $(X/Z\ni z,B)$ has an $n$-complement $(X/Z\ni z,B^+)$. Moreover, if the closure of $\Gamma$ belongs to $[0,1]\cap\mathbb Q$, then $B^+$ can be chosen so that $B^+\geq B$.

Uniform complements are a key input for canonical bundle formulas and boundedness arguments. The conjecture is known in dimensions at most three, while the general case remains open.