Logarithmic clique-width for cyclic box graphs

Let $G$ be a graph and let ${\cal P}$ be a monotone partition of $V(G)$ into cliques. Assume that the box graph $b(G;{\cal P})$ is a chordless cycle. The logarithmic cyclic-box conjecture. There exists a function $g$ such that

$$\operatorname{cwd}(G)\leq g(k)\log |V(G)|.$$

The preceding forest result shows a constant clique-width bound when the box graph is acyclic, while this conjecture predicts a logarithmic bound when it is a single chordless cycle. The supplied text gives no resolution, so the conjecture remains open.