Bourgain-type maximal sum-free set conjecture

Let $A$ be a set of $N$ positive integers. A subset $\Omega\subseteq\mathbb{R}/\mathbb{Z}$ is maximal sum-free if it is sum-free and has measure $1/3$. Let $\mathbb{1}_{\Omega}$ denote its characteristic function. Maximal sum-free set conjecture. There is a function $\omega(N)\to\infty$ as $N\to\infty$ such that for every set $A$ of $N$ positive integers, there exists a maximal sum-free set $\Omega\subseteq\mathbb{R}/\mathbb{Z}$ satisfying

$$\max_{x\in\mathbb{R}/\mathbb{Z}}\sum_{n\in A}\left(\mathbb{1}_{\Omega}-\frac{1}{3}\right)(nx)=\omega(N).$$

The source states that this conjecture would imply the preceding conjecture. Existing estimates give only bounded improvements, such as the $(N+2)/3$ lower bound attributed to Bourgain, so the asserted unbounded improvement remains open.