Polyhedrality conjecture for the modular cone

Let $\cone_k$ denote the modular cone associated with weight $k$. Polyhedrality conjecture. Suppose that $k\equiv 2$ modulo $4$ and $k>4$. The cone $\cone_k$ is polyhedral. The paper proves a sufficient condition for polyhedrality and verifies it computationally for $k\leq 38$, while empirical evidence also supports the conjecture for larger weights.