Higher topological complexity of hyperbolic groups

We prove for non-elementary torsion-free hyperbolic groups $\Gamma$ and all
$r\ge 2$ that the higher topological complexity ${\sf{TC}}_r(\Gamma)$ is equal
to $r\cdot \mathrm{cd}(\Gamma)$. In particular, hyperbolic groups satisfy the
rationality conjecture on the $\sf{TC}$-generating function, giving an
affirmative answer to a question of Farber and Oprea. More generally, we
consider certain toral relatively hyperbolic groups.