Low shrubdepth coloring conjecture for powers of nowhere dense classes

Let $\mathcal{C}$ be a nowhere dense class of graphs, let $d$ be a positive integer, and let $\varepsilon$ be a positive real. For a graph $G$, write $G^d$ for its $d$th power, and let shrubdepth denote the structural parameter used for dense graph classes. Low shrubdepth coloring conjecture. There is a function $f\colon \mathbb{N}\to\mathbb{N}$ such that, for every natural number $p$ and every $n$-vertex graph $G\in\mathcal{C}^d$, $G$ has a coloring with $\mathcal{O}(n^{\varepsilon})$ colors in which every subgraph induced by a subset of $p$ colors has shrubdepth at most $f(p)$. This conjecture seeks to extend the corresponding low shrubdepth coloring theorem from bounded expansion classes to powers of nowhere dense classes; the cited arguments for bounded expansion do not generalize to the nowhere dense setting, and the conjecture remains open.