Original Burr–Erdős conjecture for pairs of bounded-arboricity graphs

For graphs $H_1$ and $H_2$, let $r(H_1,H_2)$ be the least integer $n$ such that every red-blue edge-coloring of $K_n$ contains a red copy of $H_1$ or a blue copy of $H_2$. The arboricity of a graph is the minimum number of forests into which its edge set can be partitioned. Original Burr–Erdős conjecture. For every natural number $d$, there exists a constant $c$ such that, for every pair of graphs $H_1$ and $H_2$ each having arboricity at most $d$,

$$r(H_1,H_2)\leq c\bigl(|V(H_1)|+|V(H_2)|\bigr).$$

This is the original pair-version formulation underlying the degenerate-graph perspective. The source presents it as the original conjecture and does not give a resolution here.