The linear induced-subgraph conjecture for circle graphs

A circle graph is the intersection graph of chords of a circle. Let $r$ and $n$ be positive integers, and let the circle graph have $n$ vertices. The linear induced-subgraph conjecture. Every $K_r$-free circle graph on $n$ vertices contains an induced subgraph on $\Omega(n)$ vertices which is $K_{r-1}$-free. This would improve the currently described bound of $\Omega(n/\log r)$, obtained from known chromatic-number estimates for circle graphs. Whether the linear bound holds is left open.