The bounded clique-width conjecture for 2-well-quasi-ordered hereditary graph classes

Let $\mathcal{C}$ be a hereditary class of finite graphs. The bounded clique-width conjecture. If $\mathcal{C}$ is $2$-well-quasi-ordered, then $\mathcal{C}$ has bounded clique-width. The paper presents this as an open conjecture, motivated by the belief that bounded clique-width is sufficient to understand the corresponding well-quasi-ordering questions.