Monomial equivalence of two six-dimensional toric codes over the field with eight elements

Let $P_6^{(4)}$ and $P_6^{(5)}$ be the polytopes indexing the corresponding six-dimensional toric codes $C_{P_6^{(4)}}$ and $C_{P_6^{(5)}}$ over $\mathbb{F}_8$. The conjecture. The codes $C_{P_6^{(4)}}$ and $C_{P_6^{(5)}}$ over $\mathbb{F}_8$ are monomially equivalent. The preceding classification establishes this equivalence in all other relevant cases except that this particular equivalence remains open; the conjecture is motivated by the fact that the two codes have exactly the same enumerator polynomial.