Reed's domination conjecture for connected cubic graphs

Let $G$ be a connected cubic graph, and let $V(G)$ denote its vertex set. A dominating set is a subset of vertices such that every vertex outside the subset has a neighbor in it. Reed's conjecture. Every connected cubic graph $G$ contains a dominating set of at most

$$\left\lceil \frac{|V(G)|}{3} \right\rceil$$

vertices. This conjecture gives a universal upper bound on the domination number of connected cubic graphs; the source presents it as a statement that has been suggested and widely discussed, without supplying evidence of a resolution.