Superlinear CZ-distance for circle graphs

Let $G$ be an $n$-vertex circle graph, and let $\operatorname{CZ}(G)$ denote its CZ-distance. Circle-graph lower-bound conjecture. There exist $n$-vertex circle graphs $G$ with

$$\operatorname{CZ}(G)=\Omega(n\log n).$$

The conjecture would close the remaining logarithmic gap between the paper's $O(n\log n)$ upper bound for circle graphs and the general linear lower bound; the source gives no resolution.