Reed's constrained colouring conjecture

Let $G$ be a graph, and let $L$ be a $(k+1)$-list assignment of $G$, meaning that each vertex $v$ has a list $L(v)$ of $k+1$ available colours. Suppose that for every vertex $v$ of $G$ and every colour $c\in L(v)$, at most $k$ neighbours $w\in N_G(v)$ satisfy $c\in L(w)$. Reed's constrained colouring conjecture. There exists a proper $L$-colouring of $G$. This conjecture strengthens ordinary list-colourability conditions by controlling the number of neighbours sharing each available colour; its status is not resolved in the supplied source.