The wedge-of-spheres conjecture for independence complexes of circle graphs

Let $G$ be a circle graph, namely the intersection graph of the chords in a chord diagram, and let $I(G)$ be its independence complex, whose simplices are the subsets of pairwise non-adjacent vertices of $G$. Wedge-of-spheres conjecture. The independence complex associated to a circle graph is homotopy equivalent to a wedge of spheres. The preceding construction shows that every wedge of spheres can arise as the homotopy type of the independence complex of a circle graph; this conjecture asserts the converse. Its status is not resolved in the supplied source.