Ravanini–Tateo–Valleriani periodicity conjecture for simply laced Y-systems

Let $X_r$ and $X'_{r'}$ be simply laced Dynkin diagrams of finite type, with index sets $I$ and $I'$, Coxeter numbers $h$ and $h'$, and ranks $r$ and $r'$. A family of positive real numbers $Y_{ii'}(u)$, for $i\in I$, $i'\in I'$, and $u\in\mathbb{Z}$, satisfies the Y-system $\mathbb{Y}(X_r,X'_{r'})$ when

$$Y_{ii'}(u-1)Y_{ii'}(u+1)=\frac{\displaystyle\prod_{j:j\sim i}(1+Y_{ji'}(u))}{\displaystyle\prod_{j':j'\sim i'}(1+Y_{ij'}(u)^{-1})},$$

where adjacency is taken in the respective Dynkin diagrams. Ravanini–Tateo–Valleriani periodicity conjecture. Every such solution satisfies

$$Y_{ii'}(u+2(h+h'))=Y_{ii'}(u).$$

The conjecture generalizes Zamolodchikov’s periodicity conjecture. The source records proofs in several special cases and states that Keller subsequently proved it in full generality.