a-n-complements conjecture

Let $a$ be a nonnegative real number. An $a$-lc over $o$ $\mathbb R$-complement of $(X/Z\ni o,B)$ is assumed in the cited complement theorems. An $a$-$n$-complement is an $n$-complement $(X/Z,B^+)$ such that $(X,B^+)$ is $a$-lc over $o$.

$a$-$n$-complements conjecture. Under the additional assumption that $(X/Z\ni o,B)$ has an $a$-lc over $o$ $\mathbb R$-complement, the cited complement theorems should yield an $a$-$n$-complement $(X/Z,B^+)$ with $n\in\mathcal N$. The set of complementary indices $\mathcal N$ may depend on $a$; restrictions on complementary indices are expected to hold for $I$, but not for approximations when $a$ is irrational. The same is expected for Alexeev pairs of index $m$.

This is an $a$-lc strengthening of the existence and boundedness theory for complements. It remains open in the generality stated, and the source notes that the dependence on irrational $a$ requires weaker conclusions for approximations.