The Fourier–Hahn–Lévy conjecture for tractable integer-valued moments

Let $f:\mathbb{Z}\to\mathbb{Z}$ be tractable, meaning that its $f$-moment can be estimated to within a constant factor with a small sketch. The $f$-moment is

$$\sum_{v\in[n]} f(\mathbf{x}(v)).$$

Fourier–Hahn–Lévy conjecture. If $f:\mathbb{Z}\to\mathbb{Z}$ is tractable, then the $f$-moment can be estimated to within a constant factor with a Lévy-Tower, either directly or after applying the Fourier–Hahn–Lévy method. The conjecture asserts that this method captures all tractable integer-valued $f$-moments; the source does not provide a resolution.