Two-line intersection conjecture for rich-line-free point sets

Let $P\subseteq\mathbb{R}^2$ be a finite point set. For a line $l$, let $L(P)\cap l$ denote the set of points where lines from $L(P)$ intersect $l$. Assume that no more than $|P|^{1-\delta}$ points of $P$ lie on any line, and let $l_1$ and $l_2$ be distinct arbitrary lines. For every $\delta>0$ there exists $\epsilon=\epsilon(\delta)>0$ such that

$$|L(P)\cap l_1|+|L(P)\cap l_2|\gg |P|^{1+\epsilon}.$$

Two-line intersection conjecture. For all $\delta>0$, there exists $\epsilon=\epsilon(\delta)>0$ for which the displayed bound holds for every such $P$, $l_1$, and $l_2$. This conjecture predicts a sum-product-type expansion for intersections of the line set with two distinct lines, once very rich lines in $P$ are excluded; its status is open in the supplied source.