The inverse Littlewood conjecture for structure of low-norm sets

Let $A\subset\mathbb{Z}$ be a finite set of size $N$, and let $K>0$. Write $1_A$ for its indicator and define

$$\widehat{1_A}(\theta)=\sum_{n\in A}e(-n\theta),\qquad e(x)=e^{2\pi i x}.$$

A finite arithmetic progression is a finite subset of $\mathbb{Z}$ with constant nonzero common difference. The inverse Littlewood conjecture. If

$$\\|\widehat{1_A}\\|_1\leqslant K\log N,$$

then there are integers $m,r\ll_K1$, finite arithmetic progressions $P_1,\ldots,P_m\subset\mathbb{Z}$ of length at most $N$, finite sets $X_1,\ldots,X_r\subset\mathbb{Z}$ with

$$\left|\bigcup_jX_j\right|=o_K(N),$$

and $\epsilon_1,\ldots,\epsilon_m,\eta_1,\ldots,\eta_r\in\\{-1,1\\}$ such that

$$1_A=\sum_{1\leqslant i\leqslant m}\epsilon_i1_{P_i}+\sum_{1\leqslant j\leqslant r}\eta_j1_{X_j}.$$

This is a proposed structural inverse theorem: it would describe every set with nearly minimal Fourier $L^1$ norm using boundedly many arithmetic progressions, up to a small exceptional union. No counterexample is known; analogous finite-field results are known, and the source notes a related, slightly stronger conjecture of Petridis. The conjecture would in particular imply that $A$ has dense intersection with an arithmetic progression of length at most $N$.