Homomorphism-Distinguishing Closure of Bounded Cliquewidth

Determine whether the class of graphs of cliquewidth at most k is homomorphism distinguishing closed for every fixed k.

Background

The paper relates dense Weisfeiler--Leman equivalence to homomorphism indistinguishability over bounded-cliquewidth graphs and discusses collapse results on biclique-free inputs. It leaves open whether each bounded-cliquewidth class has the stronger homomorphism-distinguishing-closure property studied in the cited literature.