Nilsequence approximation conjecture for 3-term progression dual functions

For $A\subseteq[N]$, define the dual function counting normalized 3-term arithmetic progressions with common difference $d$ by

$$F_A(d):=\mathbb{E}_x1_A(x)1_A(x+d)1_A(x+2d).$$

A 2-step nilsequence is understood in the sense of the paper, and its complexity is measured as in the paper. Nilsequence approximation conjecture. For every $\varepsilon>0$, there exists a set of 2-step nilsequences $\{\phi_j\}$ of complexity $O_\varepsilon(1)$ such that, for every $A$, there are coefficients $c_{A,j}$ and an error function $e_A$ satisfying

$$\sum_j|c_{A,j}|\leq 1,\qquad |e_A(d)|\leq\varepsilon\quad(d=1,\ldots,N),$$

and

$$F_A(d)=\sum_jc_{A,j}\phi_j(d)+e_A(d).$$

The conjecture seeks a uniform nilsequence model for the functions governing 3-term progressions with restricted common differences, analogous to the role of Fourier analysis for 2-term progressions. The source relates it to Problem 1 of Frantzikinakis and uses it as a proposed route toward the random-difference problem; no resolution is supplied.