Berry–Esseen dependency-graph conjecture

Let $(Y_k)_{k otin V}$ be a family of $N$ random variables admitting a dependency graph of degree $D\in\mathbb{Z}_{\geq 0}$. Assume that there \exists $\delta\in(2,3]$ such that $\mathcal{A}_\delta<\infty$, where $\mathcal{A}_\delta=\sum_{k\in V}\mathbb{E}[|Y_k-c_k|^\delta]$, and let $W$ be the standardized \sum and $\xi_\delta$ the renormalized standard deviation defined by

$$\xi_\delta=\left(\frac{N}{\mathcal{A}_\delta}\right)^{1/\delta}\sqrt{\frac{\operatorname{Var}(S)}{N(D+1)}}.$$

Berry–Esseen dependency-graph conjecture. The bound

$$d_{\mathrm{Kol}}(W,\mathcal{N}(0,1))\lesssim \xi_\delta^{-\delta}\left(\frac{D+1}{N}\right)^{(\delta-2)/2}$$

should hold.

This would extend the finite-moment Berry–Esseen estimate from independent variables to variables with a sparse dependency graph, with the effective sample-size factor $N/(D+1)$. The paper presents this as a conjecture; no resolution is supplied in the given text.