Graph-product structure conjecture for $k$-nearest-neighbour graphs

A finite set $P\subset\mathbb{R}^d$ defines a $k$-nearest-neighbour graph by joining two points whenever one is among the $k$ points closest to the other. Here, $H\boxtimes\mathbb{Z}^{d-1}$ denotes the strong product of a graph $H$ and the $(d-1)$-dimensional integer lattice, and $f(k,d)$ is a function of $k$ and $d$. Graph-product structure conjecture. Every $k$-nearest-neighbour graph in $\mathbb{R}^d$ is a subgraph of

$$H\boxtimes\mathbb{Z}^{d-1}$$

for some graph $H$ with treewidth at most $f(k,d)$. This would describe these geometric graphs through a strong product with a fixed-dimensional lattice and a factor of bounded treewidth, complementing known separator results for $k$-nearest-neighbour graphs. The source poses this as a question and gives no resolution.