Parity conjecture for critical points of the circulant analytic functional

For $N=4$, decompose the functional into components $\Phi=\Phi_0+\Phi_1+\Phi_2+\Phi_3$, where the $\Phi_i$ are the components defined in the preceding formulas. Parity conjecture for critical points. At a critical point, the value of $\Phi_i$ depends only on the parity of $i$. This conjecture was found by computer and concerns a structural property of the critical-point values; the source does not provide a proof or resolution.