Generalized forbidden out-degree conjecture

Let $G$ be a loopless graph, and let $d(v)$ denote the degree of each vertex $v\in V(G)$. Given a function $F:V(G)\to 2^{\mathbb N}$, an orientation is $F$-avoiding if its out-degree $d_D^+(v)$ satisfies $d_D^+(v)\notin F(v)$ for every $v\in V(G)$. Generalized forbidden out-degree conjecture. If $|F(v)|<\tfrac{1}{2}d(v)$ for all $v\in V(G)$, then $G$ admits an $F$-avoiding orientation. This generalization is motivated by allowing the degree and forbidden list to vary from vertex to vertex. The source verifies it for some new cases, including 2-degenerate graphs and forbidden sets with specific structure, but the full statement remains open.