Polynomial-time homotopy-type conjecture for independence complexes of circle graphs

Let $G$ be a circle graph, and let $I(G)$ be its independence complex. The size of the input is measured by the number of vertices of $G$. Polynomial-time homotopy-type conjecture. The homotopy type of $I(G)$ can be found in polynomial time with respect to the number of vertices of $G$.

This algorithmic conjecture is closely related to Adamaszek's question and would imply polynomial-time determination of the extreme Khovanov spectrum in the relevant setting. The paper establishes polynomial-time computation for independence complexes associated with $4$-braid diagrams, while the general circle-graph problem remains open.