Higher-dimensional doubly ruled incidence conjecture for lines

Let $\frak L$ be a set of $L$ lines in $\mathbb{C}^n$, and let $P_2(\frak L)$ denote the set of points incident to at least two lines of $\frak L$. Higher-dimensional doubly ruled incidence conjecture. Either

$$|P_2(\frak L)|\leq C(n)L^{\frac{n}{n-1}},$$

or there exist an integer $m$ with $2\leq m\leq n-1$ and a generically double-ruled affine variety $Z$ of dimension $m$ containing at least $L^{\frac{m-1}{n-1}}$ lines of $\frak L$. Here an affine variety is irreducible by definition. The conjecture is motivated by the difficulty of extending ruled-surface methods to higher dimensions, where doubly ruled varieties can have every dimension between $2$ and $n-1$; the source says that proving it may require a generalization of those methods.