The inverse Littlewood conjecture for dissociated dimension

Let $A\subset\mathbb{Z}$ be a finite set of size $N$, and define $\dim A$ to be the size of the largest dissociated subset of $A$. Let $K>0$. The inverse Littlewood dimension conjecture. If $N$ is sufficiently large depending only on $K$ and

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

then

$$\dim A\ll_K(\log N)^2.$$

Furthermore, $A$ contains a subset of size $\gg_KN$ and dimension $\ll_K\log N$. This would sharpen the known upper bound $\dim A\ll_K(\log N)^3$ and match, up to constants, examples formed from a large arithmetic progression together with a dissociated set of size about $(\log N)^2$.