Critical-point conjecture for multireference alignment loss landscapes

Let $x\in\mathcal{M}_{m_2}$ be an odd-size signal and let $\tau>0$. Write $\operatorname{supp}(\widehat{m}_2)$ for the support of the Fourier transform of the second moment, and identify critical points up to circular shifts. Critical-point conjecture. The loss function \mathcal{L}(\begin{tikzpicture}\draw[fill=black](0,0)++(0,0.2em) circle (1.2 pt);\end{tikzpicture};x,\tau) has exactly

$$2^{\#\operatorname{supp}(\widehat{m}_2)}$$

critical points up to circular shifts, namely those described in Proposition. In particular, $x$ and $-x$ are respectively the global minimum and maximum of \mathcal{L}(\begin{tikzpicture}\draw[fill=black](0,0)++(0,0.2em) circle (1.2 pt);\end{tikzpicture};x,\tau), while all remaining critical points are saddle points. This conjecture concerns the critical-point structure and optimization landscape of the multireference alignment loss for odd signal length; the supplied text does not indicate that it has been proved or disproved.