Characterisation of polynomial-growth graphs by products of trees

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 has polynomial growth with parameters $c$ and $d$ when $f_G(r)\leqslant cr^d$, and each $T_i$ below is a tree with growth bounded linearly in $r$.

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

$$T_1\boxtimes\cdots\boxtimes T_d\boxtimes K_{g(c,d)},$$

where each $T_i$ is a tree of growth $f_{T_i}(r)\leqslant h(c,d)r$.

This is presented as a more general rough characterisation of graphs of polynomial growth, extending the preceding linear-growth conjecture. Its status is unresolved in the supplied source.