The neighborhood-complexity characterization of nowhere-dense classes

Let $\mathcal{C}$ be a graph class closed under taking subgraphs. For $r\in\mathbb{N}$, a graph $G$, and $A\subseteq V(G)$, let $N_r^G[u]$ be the ball of radius $r$ around $u$, and define the $r$-neighborhood complexity of $A$ by

$$\nu_r(G,A)=\left|\{N_r^G[u]\cap A\colon u\in V(G)\}\right|.$$

The neighborhood-complexity characterization. The class $\mathcal{C}$ is nowhere dense if and only if there exists a function $f_{\mathrm{nei}}(r,\varepsilon)$ such that

$$\nu_r(G,A)\leq f_{\mathrm{nei}}(r,\varepsilon)\cdot |A|^{1+\varepsilon}$$

for all $r\in\mathbb{N}$, $\varepsilon>0$, graphs $G\in\mathcal{C}$, and vertex subsets $A\subseteq V(G)$.

Linear neighborhood complexity characterizes subgraph-closed classes of bounded expansion, and this conjecture asks for the analogous almost-linear characterization of nowhere-dense classes. The source presents it as an open problem posed by Reidl et al.