The tree product conjecture for graphs of polynomial growth

Let $H$ be a finite graph, and write $f_H(r)$ for its growth function, namely the maximum number of vertices in a ball of radius $r$. For graphs $G_1,\dots,G_k$, let $G_1 \boxtimes \dots \boxtimes G_k$ denote their strong product, and let $K_m$ denote the complete graph on $m$ vertices. A graph class has degree-$d$ polynomial growth if its growth is bounded by $O(r^d)$. The tree product conjecture. There exist functions $g \colon \mathbb{R} \times \mathbb{N} \to \mathbb{N}$ and $h \colon \mathbb{R} \times \mathbb{N} \to \mathbb{N}$ such that for every $c\in \mathbb{R}_{\geqslant 1}$ and $d \in \mathbb{N}$, every finite graph $H$ with growth $f_H(r)\leqslant cr^d$ is isomorphic to a subgraph of

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

where each $T_i$ is a tree with growth $f_{T_i}(r)\leqslant h(c,d)r$. The conjecture seeks the tight number $d$ of product factors for graphs of degree-$d$ polynomial growth, with each factor a tree of linear growth and with a bounded clique factor; its resolution would strengthen the known product-structure theorem for graph classes of linear growth.