The connected-subgraph density conjecture for r-colorings

Let $K_\NN$ be the complete graph on the positive integers. For a subgraph $T$ of $K_\NN$, write $\bar d(T)$ for the upper density of its vertex set and $\bar d_S(T)$ for its strong upper density; in particular, $T$ is connected. Connected-subgraph density conjecture. For $r\geq4$, every $r$-coloring of $K_\NN$ contains a monochromatic connected subgraph $T$ such that

$$\bar d(T)\geq\bar d_S(T)\geq\frac{1}{r-1}.$$

This extends the infinite analogue of Gyárfás's finite connected-subgraph theorem, proved in the paper for $r\in\{2,3\}$, and the bound is best possible when $r-1$ is a prime power.