Symmetry conjecture for minimizers of the circulant analytic functional

Let $N=4n$, and let

$$\Phi=\sum_{i+j+k+l=0}q_iq_jq_kq_l$$

be minimized over vectors $q\in\mathbb T^N$ satisfying $\bar q_i=q_{-i}$. In the parametrization described immediately before the conjecture, write the corresponding vectors as having parameters $a$ and $c$. Symmetry conjecture for minimizers. The minimum over all such vectors is equal to the minimum over the vectors in this parametrized family satisfying $a=c$. The computations at $N=8$ and $N=12$ motivate this symmetry reduction, but the source does not establish it in general.