The doubly extended exchange-complex homotopy conjecture

Let $\mathcal{A}$ be a doubly extended cluster algebra of rank $n+2$. Its exchange complex is the complex associated with the exchange structure, and its doubly extended associahedron is the associated dual quotient complex. The doubly extended exchange-complex homotopy conjecture. The exchange complex of $\mathcal{A}$ is homotopy equivalent to $S^{n-1}$. The doubly extended associahedron associated with $\mathcal{A}$ is homotopy equivalent to $S^{n-1}\times S^2$ in all cases other than $E_8^{(1,1)}$, where it instead is homomorphic to $S^7\times S^1\times S^1$. The paper explicitly notes that doubly extended associahedra are not generally homotopy equivalent to spheres; the stated product decompositions remain conjectural.