The 8/9-density conjecture for monochromatic paths

Let $K_\NN$ be the complete graph on the positive integers, and define the upper density of a vertex set $A\subseteq\NN$ by $\bar d(A)=\limsup_{n\to\infty}|A\cap[n]|/n$. The upper density of a path is the upper density of its vertex set. The 8/9-density conjecture. Every $2$-coloring of $K_\NN$ contains a monochromatic path of upper density at least $8/9$. The bound would be best possible by the example cited in the source.