Twin-width essentiality beyond permutation graphs

Determine which graphs other than permutation graphs are essential with respect to twin-width; specifically, establish whether there exists any graph besides permutation graphs for which a hereditary class of unbounded twin-width has bounded twin-width after restricting to graphs that are free of that graph.

Background

The paper defines a graph as essential for a parameter if it belongs to a hereditary obstruction whose ambient class has unbounded parameter value while the obstruction-free graphs have bounded parameter value. For treewidth, the paper proves that every graph is essential, and it observes an analogous result for twin-width for every permutation graph because permutation graphs form a minimal hereditary class of unbounded twin-width.

The authors state that, apart from permutation graphs, no other graph is known to be essential for twin-width. This leaves open the classification of graphs that can force the transition from unbounded to bounded twin-width in hereditary graph classes.