The non-3-colorability conjecture for countable undirected multigraphs

An undirected countable multigraph is a countable undirected graph whose edges have weights in $\mathbb{N}$. A majority coloring is a coloring such that, at every vertex, the total weight of bichromatic incident edges is at least the total weight of monochromatic incident edges.

Multigraph non-3-colorability conjecture. There exists an undirected countable multigraph that is not majority 3-colorable.

Finite multigraphs are majority 2-colorable by a max-cut argument, but the directed construction in the paper does not directly transfer to multigraphs. The asserted countable example remains open in the source.