The bounded-degree monochromatic density conjecture

Let $\bar d(G)$ denote the upper density of the vertex set of a graph $G$ embedded in $K_\NN$. Bounded-degree density conjecture. For every $\Delta\in\NN$, there exists $c>0$ such that for every infinite graph $G$ with maximum degree at most $\Delta$ and every $2$-coloring of $K_\NN$, there is a monochromatic copy of $G$ with

$$\bar d(G)\geq c.$$

This is the proposed infinite analogue of the finite bounded-degree embedding theorem of Chvátal, Rödl, Szemerédi, and Trotter; it asks for a uniform positive density depending only on the maximum-degree bound.