Conjecture on compatibility minimizers and pure connected components

Let ${\bm{\tau}}\in S_n^D$, and let $\nabla({\bm{\tau}};\eta)$ be the nonnegative quantity whose vanishing characterizes the relevant compatible configurations. For a permutation or tuple of permutations, let $\Pi_\mathrm{p}$ denote its partition into pure connected components, and let $H_{{\bm{\tau}},\Pi_\mathrm{p}({\bm{\tau}})}$ be the corresponding subgroup or set defined in the source. Then compatibility-minimizer conjecture. every $\eta\in S_n$ satisfying

$$\nabla({\bm{\tau}},\eta)=0$$

satisfies

$$\Pi_\mathrm{p}(\eta)\leq\Pi_\mathrm{p}({\bm{\tau}}),$$

or equivalently, $\eta\in H_{{\bm{\tau}},\Pi_\mathrm{p}({\bm{\tau}})}$. The source also notes that this conjecture would follow from subadditivity of the Gaussian scaling function. This is a structural conjecture about the minimizers governing dominant contributions in the asymptotic expansion of unitarily invariant random tensors; the source gives no proof or resolution.