The list 2-coloring conjecture for countable acyclic digraphs

Let $D$ be a countable acyclic digraph, and assign to each vertex $v$ a list $L(v)$ of available colors. The digraph is majority $k$-choosable if, for every such assignment with $|L(v)|=k$, it has a majority coloring using a color from $L(v)$ at each vertex, where at every vertex at most half of its outgoing edges are bad. Acyclic digraph list-coloring conjecture. Every countable acyclic digraph is majority $2$-choosable. The greedy algorithm proves the analogous statement for finite acyclic digraphs, but the countable list version is left open.