Hofmeister–Joret–Wood conjecture on tree densities in minor-closed graph classes

Let $s,t$ be integers with $t\geqslant s\geqslant 1$, and let $\mathcal{B}_{s,t}$ be the class of graphs containing no $K_{s,t}$ minor. For a fixed graph $H$, define an $s$-separation of $H$ to be a pair $(A,B)$ of edge-disjoint subgraphs with $A\cup B=H$, $V(A)\setminus V(B)\neq\emptyset$, $V(B)\setminus V(A)\neq\emptyset$, and $|V(A)\cap V(B)|=s$. A $(\leqslant s)$-separation is an $s'$-separation for some $s'\leqslant s$, and two separations $(A,B)$ and $(C,D)$ are independent if $E(A)\cap E(C)=\emptyset$ and $(V(A)\setminus V(B))\cap(V(C)\setminus V(D))=\emptyset$. Define $f_s(H):=1$ if $H$ has no $(\leqslant s)$-separation; otherwise, let $f_s(H)$ be the maximum number of pairwise independent $(\leqslant s)$-separations in $H$. Hofmeister–Joret–Wood conjecture. For every fixed graph $H$ with no $K_{s,t}$ minor,

$$C(H,\mathcal{B}_{s,t},n)\in\Theta\bigl(n^{f_{s-1}(H)}\bigr).$$

This conjecture proposes the correct asymptotic order for the number of copies of an arbitrary fixed graph in graphs excluding a fixed complete bipartite minor. The paper establishes the corresponding asymptotic behaviour for forests in several sparse graph classes, while this assertion concerns general graphs and is presented as an open problem.