The nowhere-dense domination-set kernelization dichotomy

Let $\mathcal{C}$ be a class of graphs closed under taking subgraphs. A class is nowhere dense if it belongs to the sparse graph-class regime characterized by excluding arbitrarily large clique subdivisions at bounded depths; it is somewhere dense otherwise. The \textsc{Distance-$r$ Dominating Set} problem asks whether a given graph has a dominating set whose vertices are within distance $r$ of every vertex.

The nowhere-dense domination-set kernelization dichotomy. If $\mathcal{C}$ is nowhere dense, then for each $r\in\mathbb{N}$, the \textsc{Distance-$r$ Dominating Set} problem admits an almost linear kernel on $\mathcal{C}$. Otherwise, if $\mathcal{C}$ is somewhere dense, then for some $r\in\mathbb{N}$, the \textsc{Distance-$r$ Dominating Set} problem is $\mathsf{W}[2]$-hard.

The second direction was already known for somewhere dense classes closed under taking subgraphs, while the conjectured extension of almost-linear kernelization from domination radius one to every fixed radius would establish the claimed dichotomy for parameterized complexity.