Zeros-of-the-derivative conjecture for nonunique minimal sub-sums

Let

$$f(t)=\sum_j\alpha_j C_j(t)+\sum_k\beta_k C_k(t)$$

be a polynomial potential on the unit circle, and suppose that minimizing over the stated class of equally distributed measures reduces to choosing a minimal sub-sum of its coefficient sequence. Zeros-of-the-derivative conjecture. If $f(t)$ does not have a unique minimal sub-sum, the minimizer will be a measure supported on a set of points whose inner products correspond to the zeros of the derivative of $f(t)$. The conjecture addresses the nonuniqueness observed for polynomial potentials, where distinct configurations such as antipodal points and regular polygons can attain the same value; whether the asserted support description always holds remains open.