Bounded clique-width from bounded chordless-cycle subgraphs

Let $G$ be a graph, and let ${\cal P}=(A_1,\ldots,A_k)$ be a monotone partition of $V(G)$ into cliques. The box graph $b(G;{\cal P})$ has a chordless cycle $A_{i_1}\dots A_{i_c}A_{i_1}$ for which the induced subgraph on the corresponding parts has bounded clique-width. The bounded-cycle clique-width conjecture. There exists a function $f$ such that, if for every chordless cycle $A_{i_1}\dots A_{i_c}A_{i_1}$ of $b(G;{\cal P})$ one has

$$\operatorname{cwd}(G[A_{i_1}\cup\dots\cup A_{i_c}])\leq x,$$

then

$$\operatorname{cwd}(G)\leq f(k,x).$$

This would extend the forest case, where the clique-width is bounded by $k+1$, to monotone partitions whose box graphs may contain cycles. The supplied text gives no resolution, so the conjecture remains open.