The minimal-class conjecture for sparse hereditary graph classes

Let $\mathcal{C}$ be a sparse hereditary class of graphs with unbounded tree-width. A hereditary graph class is minimal of unbounded tree-width if it has unbounded tree-width, while every proper hereditary subclass has bounded tree-width. Minimal-class conjecture. The class $\mathcal{C}$ does not contain a minimal class of unbounded tree-width. The preceding results establish this for hereditary classes of bounded vertex degree, classes with an excluded minor, and the path-star hereditary classes $\mathcal{R}^{\nu}$, $\mathcal{R}^{\kappa(q)}$ and $\mathcal{R}^{\eta}$; whether it holds for all sparse hereditary classes remains open.