KMP$_2$ conjecture for arbitrary weighted hypergraphs

Let $\Gamma=([n],w)$ be an arbitrary hypergraph with non-negative weights. Let $\mathcal{KMP}_k$ be the discrete KMP process with $k$ indistinguishable particles on $\Gamma$, and let $\lambda_{\min}^*(\Gamma,\mathcal{KMP}_k)$ denote its smallest non-trivial eigenvalue. For an irreducible representation $\rho$ of $\mathrm{U}(n)$, write $\lambda_{\min}(\Gamma,\rho)$ for the smallest eigenvalue in its spectrum. KMP$_2$ conjecture.

$$\lambda_{\min}^*(\Gamma,\mathcal{KMP}_2)=\inf_{\mathrm{triv}\ne\rho\in\mathrm{Irr}(\mathrm{U}(n))}\lambda_{\min}(\Gamma,\rho)=\min_{\mathrm{triv}\ne\rho\in\mathrm{Irr}(\mathrm{U}(n))}\lambda_{\min}(\Gamma,\rho).$$

Equivalently, the spectral gap on the regular representation of $\mathrm{U}(n)$ would coincide with that of the two-particle KMP process. The source presents this as equivalent to the preceding conjecture and gives no resolution.