The maximal tubular incidence conjecture for direction-separated lines

Let $P_r$ be the set of points in $\mathbb{R}^3$ that lie in at least $r$ of the $\delta$-tubes associated with the line segments. Let $\lambda$ denote Lebesgue measure.

Maximal tubular incidence conjecture. Let ${l_i}_{i\in\{1,\dots,L\}}$ be $L$ unit line segments in $\mathbb{R}^3$ whose directions are $\delta$-separated. Then there is a constant $C>0$ such that

$$\lambda(P_r)\leq C\delta^3L^{1.5}/r^{1.5}.$$

The paper explains that this estimate is sharp for highly clustered directions and gives a stronger expected bound in a random model. Whether the displayed estimate holds under direction separation is posed as an open problem.