The strong union-of-lines conjecture

Let $\epsilon>0$ be any constant and let $q$ be a sufficiently large prime power. Let $L$ be a set of lines in $\mathbb{F}_q^3$ satisfying

$$|L|\geq q^{5/2+\epsilon}$$

and such that no plane contains more than $(1/2)q^{3/2}$ lines of $L$. Strong union-of-lines conjecture. Then

$$|P(L)|\geq q^3-O(q^{5/2}).$$

This is substantially stronger than the weak union-of-lines conjecture and would give an almost-full union of points under a denser line hypothesis. The authors state that they do not have a counterexample, but the conjecture remains open.