Boundedness of complements for normalized foliated structures

Let $d$ be a positive integer, and let $\Gamma\subset [0,1]$ be a DCC set of rational numbers whose closure is contained in $\mathbb Q$. A projective $\mathbb R$-complementary normalized foliated structure is written as $\mathfrak{B}:=(X,\mathcal{F},B)(t)$, where $X$ is of Fano type and $t\in\Gamma$. Boundedness of complements conjecture. There exists a positive integer $N$, depending only on $d$ and $\Gamma$, such that every such $\mathfrak{B}$ admits a monotonic $N$-complement $\mathfrak{B}^+=(X,\mathcal{F},B^+)(t^+)\geq\mathfrak{B}$ satisfying

$$B^+\geq B,\qquad t^+\geq t,$$

and

$$Nt^+\in\mathbb N,\qquad NB^+\text{ is an integral divisor},\qquad NK_{\mathfrak{B}^+}\sim 0.$$

The conjecture is proposed as a foliated analogue of boundedness of complements, motivated by its expected importance in K-stability and K-moduli theory. It is stated as a direction for future work; the source records no resolution.