The edge-isoperimetric conjecture for the permutahedron at size $6^{(n+1)/3}$

Let $\operatorname{Perm}(n)$ be the permutahedron graph, and let $i_k(G)$ denote the minimum edge-expansion among subsets of $G$ of size $k$. Set

$$k=6^{\frac{n+1}{3}}.$$

Edge-isoperimetric conjecture for the permutahedron.

$$i_k\bigl(\operatorname{Perm}(n)\bigr)=\frac{n+1}{3}-1.$$

This conjecture asserts that the heuristic construction using a Cartesian product of copies of the $6$-cycle gives the optimal edge expansion at this scale. The surrounding discussion does not provide a proof or a resolution.