Degree-two and degree-three recovery for symmetric cryo-EM

Let $H=\mathbb{Z}/L\mathbb{Z}\subset\mathrm{SO}(3)$ be generated by rotation through $2\pi/L$ about the $z$-axis, and consider homogeneous projected cryo-EM with $S$ shells and $F$ frequencies restricted to the $H$-fixed subspace. Symmetric cryo-EM conjecture. For every $S\geq2$, $F\geq2$, and $L\geq2$, generic list recovery is possible at degree $3$. It is possible at degree $2$ if and only if $F<L$. The statement is based on tests of small examples; when $F<L$, the signal in fact has the larger $\mathrm{SO}(2)$ symmetry.