Infinite Burr–Erdős conjecture for positive-density monochromatic copies

A graph $G$ is $d$-degenerate if it has an ordering of its vertices $v_1,v_2,\ldots$ such that

$$|N(v_i)\cap\{v_1,\ldots,v_{i-1}\}|\leq d$$

for every $i\geq 1$. The upper density of a set $V\subseteq\mathbb N$ is $\limsup_{n\to\infty}|V\cap\{1,\ldots,n\}|/n$. Infinite Burr–Erdős conjecture. For all positive integers $d$, there exists a real number $c_d>0$ such that if $G$ is a countably infinite $d$-degenerate graph with no finite dominating set, then in every $2$-coloring of the edges of $K_{\mathbb N}$, there exists a monochromatic copy of $G$ with vertex set $V\subseteq\mathbb N$ such that the upper density of $V$ is at least $c_d$. The source gives no resolution of this infinite analogue, which remains open.