Bounded cliquewidth under proper separator decompositions

Let $\mathcal C$ and $\mathcal B$ be classes of graphs closed under taking induced subgraphs. Suppose that every 2-connected graph in $\mathcal C$ is either in $\mathcal B$ or has a proper 2-separator or a proper $P_3$-separator. Assume that, for some constant $a$, every graph in $\mathcal B$ has cliquewidth at most $a$.

Cliquewidth preservation conjecture. There is a function $f$ such that every graph in $\mathcal C$ has cliquewidth at most $f(a)$.

This conjecture asks whether decomposing graphs along proper 2-separators and proper $P_3$-separators preserves bounded cliquewidth. Its status is not resolved in the supplied source.