Foliated complement conjecture

Let $(X/Z\ni z,\mathcal{F},B)$ be an $(\epsilon,\mathbb{R})$-complementary foliated triple of dimension $d$, with coefficients in a DCC set $\Gamma\subset[0,1]$. An $(\epsilon,n)$-complement is the corresponding bounded complement with index $n$.

Foliated complement conjecture. For every positive real number $\epsilon$ and positive integer $d$, there exists a positive real number $n$ depending only on $\epsilon$, $d$ and $\Gamma$ such that, if either $\epsilon=0$ or $-K_{\mathcal{F}}$ is big over $Z$, then $(X/Z\ni z,\mathcal{F},B)$ has an $(\epsilon,n)$-complement. Moreover, if $\bar\Gamma\subset\mathbb Q$, it has a monotonic $(\epsilon,n)$-complement.

This is the foliated analogue of Shokurov's boundedness of complements conjecture. It is known for ordinary pairs in several cases, including dimension two and certain Fano-type settings, while the foliated conjecture remains open in general.