  In their celebrated paper (arXiv:0906.1238), Caputo, Liggett and Richthammer
proved Aldous' conjecture and showed that for an arbitrary finite graph, the
spectral gap of the interchange process is equal to the spectral gap of the
underlying random walk. A crucial ingredient in the proof was the Octopus
Inequality - a certain inequality of operators in the group ring
$\mathbb{R}[S_n]$ of the symmetric group. Here we generalize the Octopus
Inequality and apply it to generalize the Caputo-Liggett-Richthammer Theorem to
certain hypergraphs, proving some cases of a conjecture of Caputo.
