Caputo's shuffle conjecture for weighted hypergraphs

Let $G$ be a finite weighted hypergraph with non-negative weights. At each step, choose a hyper-edge with probability proportional to its weight and then choose a uniformly random permutation of the vertices in that hyper-edge. Let $\lambda_2^{\mathrm{RW}}(G)$ and $\lambda_2^{\mathrm{IP}}(G)$ be the second-largest eigenvalues of the corresponding random-walk and interchange-process transition operators. Caputo's conjecture. For every such $G$,

$$\lambda_{2}^{\mathrm{RW}}(G)=\lambda_{2}^{\mathrm{IP}}(G).$$

This is the proposed hypergraph generalization of the Aldous--Caputo--Liggett--Richthammer theorem. The supplied source gives no resolution status for the conjecture.