The reflection-permutation conjecture for the free energy of random tensor networks

Let $X$ be the set on which the permutations act, let $\operatorname{Sym}(X)$ be its symmetric group, let $F(g)$ be the free energy associated with $g\in\operatorname{Sym}(X)$, and let $\pi$ be the reflection permutation. The parameters $q$ and $n$ specify the tensor-network model.

Reflection-permutation conjecture. The reflection permutation $\pi$ gives the true minimum for all $(q,n)$, namely

$$\min_{g\in\operatorname{Sym}(X)} F(g)=F(\pi).$$

The conjecture identifies the minimizer of the free energy with the reflection permutation. In the supplied discussion this is motivated by the analysis of the $n=2$ case and the lower bound associated with the multiway-cut conjecture, but no resolution is stated.