The periodicity conjecture for pairs of Dynkin diagrams

Let $\Delta$ and $\Delta'$ be Dynkin diagrams with vertex sets $I$ and $I'$. Let $A$ and $A'$ be their incidence matrices, so that if $C$ is the Cartan matrix of $\Delta$ and $J$ is the identity matrix of the same format, then $A=2J-C$, and similarly for $A'$. Let $h$ and $h'$ be the Coxeter numbers of $\Delta$ and $\Delta'$. The $Y$-system of algebraic equations associated with $(\Delta,\Delta')$ is the system in the variables $Y_{i,i',t}$, where $(i,i')\in I\times I'$ and $t\in\mathbb Z$, given by

$$Y_{i,i',t-1}Y_{i,i',t+1}=\frac{\prod_{j\in I}(1+Y_{j,i',t})^{a_{ij}}}{\prod_{j'\in I'}(1+Y_{i,j',t}^{-1})^{a'_{i'j'}}}.$$

The periodicity conjecture. All solutions to this system are periodic in $t$ of period dividing $2(h+h')$. This conjecture predicts a uniform periodicity phenomenon for algebraic $Y$-systems associated with pairs of Dynkin diagrams; the paper proves it using cluster algebras and their categorification via triangulated categories, so the claim is solved.