Logarithmic clique-width conjecture for classes of bounded stretch-width

Let $\mathcal C$ be a class of graphs of bounded stretch-width. For an $n$-vertex graph $G\in\mathcal C$, let the clique-width of $G$ be the minimum number of labels needed to construct $G$ using the clique-width operations. Logarithmic clique-width conjecture. There is a constant $c$ such that for every $n$-vertex graph $G\in\mathcal C$, the clique-width of $G$ is at most $c\log n$. This would clarify whether bounded stretch-width generally implies essentially logarithmic clique-width; the paper provides a separating example with essentially logarithmic clique-width, but the general statement is posed as an open question.