The weak union-of-lines conjecture

Let $L$ be a set of lines in $\mathbb{F}_q^3$, and define the union of lines by

$$P(L)=\bigcup_{\ell\in L}\{p\mid p\in\ell\}.$$

Weak union-of-lines conjecture. If

$$|L|=\Omega(q^3)$$

and no plane contains $\omega(q)$ lines of $L$, then

$$|P(L)|\geq(1-o(1))q^3.$$

This conjecture would imply the three-dimensional case of the Nikodym set density conjecture. The paper proves a weaker bound and does not resolve this statement.