Green–Konyagin–Littlewood conjecture for the Fourier algebra norm

Let $p$ be a prime number and let $A\subset\mathbb{Z}/p\mathbb{Z}$ have density bounded away from $0$ and $1$ by an absolute constant. Green–Konyagin–Littlewood conjecture. Then

$$\|\chi_A\|_{A(\mathbb{Z}/p\mathbb{Z})}\gg\log p.$$

This strengthens the lower bound proved by Green and Konyagin, replacing an order of $(\log p/\log\log p)^{1/3}$ with the conjectured logarithmic growth. The source presents the claim as a conjecture and gives no resolution.