Zamolodchikov–Ravanini–Tateo–Valleriani periodicity and dilogarithm conjecture for Y-systems

Let $(X,X')$ be simply-laced Dynkin diagrams of finite type, with vertex sets $I$ and $I'$, ranks $r,r'$, and Coxeter numbers $h,h'$. Let $Y_{a,a'}(u)\in\mathbb{R}_{>0}$ for $(a,a')\in I\times I'$ and $u\in\mathbb{Z}$ be any positive real solution of the $Y$-system of type $(X,X')$:

$$Y_{a,a'}(u+1)Y_{a,a'}(u-1)=\frac{\prod_{b\in I:\,b\sim a}(1+Y_{b,a'}(u))}{\prod_{b'\in I':\,b'\sim a'}(1+Y_{a,b'}(u)^{-1})}.$$

Zamolodchikov–Ravanini–Tateo–Valleriani conjecture. The solution is periodic with

$$Y_{a,a'}(u+2(h+h'))=Y_{a,a'}(u),$$

and satisfies the dilogarithm identity

$$\sum_{u=0}^{2(h+h')-1}\sum_{(a,a')\in I\times I'}\tilde{L}(Y_{a,a'}(u))=2rr'h\frac{\pi^2}{6}.$$

These claims generalize the constant-system identities and were motivated by integrable mathematical-physics models. The supplied text says that both parts were later proved in full generality using cluster-algebra structures behind the $Y$-systems.