2 problems

Sphericity conjecture. Provided $N$ is non-negative, the arc complex $Arc(F)$ is PL-homeomorphic to the sphere of dimension $N$.

The convex-polytope conjecture. The map $\Phi_h\colon T_c(S)\times\mathbb{R}_{>0}^V\longrightarrow\mathbb{R}^E$ is a smooth embedding, and its image is a convex polytope.