Unfriendly Partition Conjecture for countably infinite graphs

A majority $2$-vertex-coloring of a graph is a coloring of its vertices with two colors such that, at every vertex, the number of neighbors having the same color is at most the number having the other color. Unfriendly Partition Conjecture. Every countably infinite graph admits a majority $2$-vertex-coloring. This is the countably infinite restriction of the broader Cowan–Emerson conjecture. Every infinite graph admits a majority $3$-vertex-coloring, but whether every countably infinite graph admits a majority $2$-vertex-coloring remains open.