Shakan's stronger conjecture on the maximum of the sum-free Fourier function

Let $A\subset\mathbb{Z}_{>0}$ be a set of size $n$, and let $S$ be a positive integer. Define

$$f(x)=1_{[1/3,\,2/3)}-1/3,$$ $$f_A(x)=\sum_{a\in A}f(ax),$$

and

$$m_A=\max_{x\in\mathbb{R}}f_A(x).$$

Shakan's stronger conjecture. For every positive integer $S$, there exists $N_S\in\mathbb{Z}$ such that

$$m_A\geq\frac{S}{3}$$

for every $n\geq N_S$.

This strengthens the conjecture on arbitrarily large improvements for sum-free subsets by seeking the improvement through the maximum of the explicitly defined function $f_A$. Bourgain's earlier estimates answer the weaker conjecture for typical sets, while this stronger assertion remains open.