The LCW Enumeration Conjecture for permutation classes

A permutation class $\mathcal{C}$ is broadly rational if every finitely based subclass of $\mathcal{C}$ has a rational generating function, and it is strongly rational if it and all its subclasses have rational generating functions. Let $G_\mathcal{C}$ be the graph class associated with $\mathcal{C}$ by the permutation-graph construction, and let $\operatorname{wqo}$ denote well-quasi-ordering. The LCW Enumeration Conjecture. A permutation class $\mathcal{C}$ is broadly rational if and only if $G_\mathcal{C}$ has bounded linear clique-width. Moreover, $\mathcal{C}$ is strongly rational if and only if $G_\mathcal{C}$ has bounded linear clique-width and is well-quasi-ordered. This conjecture would connect enumeration properties of permutation classes with linear clique-width; the source presents it as an open unifying framework and lists several consequences that are themselves known conjectures.