Gajarsky–Pilipczuk–Toruńczyk conjecture on cliquewidth obstruction classes via walls

For a graph class $\mathscr{C}$, write $\mathscr{C}$ transduces a class $\mathscr{D}$ when the class is obtainable by the relevant first-order transduction. A graph class has unbounded cliquewidth when its cliquewidth is not bounded.

Gajarsky–Pilipczuk–Toruńczyk's cliquewidth obstruction conjecture. A class of graphs $\mathscr{C}$ has unbounded cliquewidth if and only if $\mathscr{C}$ transduces a class $\mathscr{D}$ that contains some subdivision of every wall.

Together with the corresponding linear-cliquewidth statement, this conjecture would characterize bounded cliquewidth through forbidden transduction patterns. The survey presents it as open.