Characterisation of graphs of linear growth by tree blow-ups

For a graph $G$, write $f_G(r)$ for its growth function, and let $G\boxtimes H$ denote the strong product of graphs $G$ and $H$. Let $K_t$ be the complete graph on $t$ vertices. A graph is said to have linear growth with parameter $c$ when $f_G(r)\leqslant cr$ for $c\ba1$.

Linear-growth product-structure conjecture. There exist functions $g\colon\mathbb{R}\to\mathbb{N}$ and $h\colon\mathbb{R}\to\mathbb{R}$ such that for any $c\geqslant 1$, every graph $G$ with growth $f_G(r)\leqslant cr$ is isomorphic to a subgraph of

$$T\boxtimes K_{g(c)}$$

for some tree $T$ with growth $f_T(r)\leqslant h(c)r$.

If true, this would approximately characterise graphs of linear growth: every subgraph of such a product has linear growth, while the preceding theorem gives a product-structure result with a possibly exponentially growing tree. The conjecture remains unresolved in the supplied source.