The classification conjecture for winning hat games on cycles with a vertex of hatness 2

Let $G$ be a cycle $C_n$ and let $h$ be a hat function such that $2 \leq h(v) \leq 4$ for every vertex $v$. Let $A\in V(G)$ be such that $h(A)=2$. The game $\mathcal{G}=\langle G,h\rangle$ is winning only in the following cases: $n=3$; there is one more vertex besides $A$ with hatness $2$; both neighbors of $A$ have hatness $3$; or one neighbor of $A$ and the vertex following that neighbor have hatness $3$. The classification conjecture. These are the only cases in which the game $\mathcal{G}$ is winning. The preceding theorem proves that every listed case is winning; the conjectural assertion is that no other hat function on a cycle satisfying the stated bounds yields a winning game.