The growth-rate conjecture for sets with few subset sums in dimension three

Let $f_3(n,2)$ denote the extremal quantity defined in the paper for dimension $3$ and parameter $K=2$. The preceding discussion gives the bounds

$$n^4\ll f_3(n,2)\ll n^5.$$

Growth-rate conjecture. We have

$$f_3(n,2)=\Theta(n^{5}).$$

This asserts that the known upper bound is sharp in the remaining case $K=d-1$ when $d=3$, corresponding to the no-three-in-a-line problem. The conjecture is motivated by the absence of better constructions; the supplied text gives no resolution.