Expression-Sparse Representation of Excluded-Biclique Classes

Determine whether every graph class of bounded $\mathcal P^\circ$-clique-width that excludes a bi-induced $K_{t,t}$ for some t has bounded expression-sparse $\mathcal P^\circ$-clique-width.

Background

Expression-sparse $\mathcal H$-clique-width requires an $(H,k)$-expression whose deparameterization is $K_{k,k}$-free, equivalently yielding bounded tree-width in the deparameterized value. The question asks whether bounded parameterized clique-width together with exclusion of a fixed bi-induced complete bipartite graph forces the existence of such sparse expressions. The authors explicitly note that they are unsure whether the answer is affirmative or negative.