Aldous–Caputo-type conjecture for the spectral gap of $\mathrm{U}(n)$

Let $\Gamma=([n],w)$ be an arbitrary hypergraph with non-negative weights. For each irreducible representation $\rho$ of $\mathrm{U}(n)$, let $\lambda_{\min}(\Gamma,\rho)$ denote the smallest eigenvalue in its spectrum, and exclude the trivial representation. Aldous–Caputo-type conjecture. The smallest non-trivial eigenvalue of the $\mathrm{U}(n)$-spectrum of $\Gamma$ is attained in one of the irreducible representations $(1,0,\ldots,0,-1)$ or $(2,0,\ldots,0,-2)$. The conjecture extends the preceding theorem from hypergraphs of restricted support to arbitrary weighted hypergraphs; the source reports supporting results and computer simulations, but does not state a resolution.