Degree-three invariant recovery in the cryo-EM model

Consider the cryo-EM model

$$y_i=P(g_i\boldsymbol{\cdot} f)+\varepsilon_i,$$

where $P$ is the tomographic projection and $R$ denotes the number of spherical shells.

Cryo-EM sample-complexity conjecture. If $R\geq 3$ is large enough, then the minimal number of observations required for accurate recovery of the $\operatorname{O}(n)$ orbit of $f$, regardless of any specific algorithm, satisfies

$$m/\sigma^6\to\infty.$$

This conjecture concerns recovery from degree-three invariants in cryo-EM. It was recently proved, so the conjectural statement is no longer open.