# Does every graph of clique-width 3 have a large induced subgraph of clique-width 2?

Is there a constant $$\alpha>0$$ such that every graph $$G$$ of clique-width $$3$$ and order $$n$$ has an induced subgraph of order at least $$\alpha n$$ and clique-width at most $$2$$ (in other words, the induced subgraph is a cograph)?

The best upper-bound I currently have on $$\alpha$$ is $$3/5$$, given by the cycle on five vertices $$C_5$$.