Shakan's conjecture on arbitrarily large improvements for sum-free subsets

Let $A\subset\mathbb{Z}_{>0}$ be a set of size $n$. A subset of $A$ is sum-free if it contains no $x,y,z$ satisfying

$$x+y=z.$$

Let $s(A)$ denote the largest size of a sum-free subset of $A$.

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

$$s(A)\geq\frac{n+S}{3}$$

for every $n\geq N_S$.

The question asks whether the standard lower bound for the largest sum-free subset can be improved by an arbitrarily large additive constant for sufficiently large sets. Bourgain proved a fixed improvement in broad circumstances, including the bound $s(A)\geq(n+2)/3$ for sets with coprime elements apart from $A=\{1,2\}$, but the asserted arbitrary improvement remains open.