Closure-size conjecture for three-dimensional line-sparse sets

Let $[n]=\{1,\ldots,n\}$, and let $S\subseteq[n]^3$ be a set that meets every line at most once. Write $\bar{S}$ for the closure of $S$ under the closure operation considered in the paper.

Closure-size conjecture. There are constants $c_1,c_2>0$ such that, if

$$|S|\geq\frac{n^2}{(\log\log n)^{c_1}},$$

then

$$|\bar{S}|\geq\frac{n^3}{(\log\log n)^{c_2}}.$$

Non-trivial lower bounds on the closure size would improve the lower bounds on ${\chi}_3(n,n)$; the surrounding discussion indicates that this remains open, with only weak bounds known for higher dimensions.