The multiplicity version of the joints problem

Let $k$ be a field and let ${\mathcal L}$ be a finite set of $L$ lines in $k^n$, with $n\geq 3$. A joint is a point $x\in k^n$ through which pass at least $n$ lines of ${\mathcal L}$ whose direction vectors are linearly independent. Write ${\mathscr J}$ for the set of joints, and let $N(x)$ be the number of $n$-tuples of lines in ${\mathcal L}$ that form a joint at $x$.

Multiplicity joints conjecture. The number of joints counted with multiplicities satisfies

$$\sum_{x\in {\mathscr J}}N(x)^{\frac{1}{n-1}}\leq c_nL^{\frac{n}{n-1}},$$

where $c_n$ is a constant depending only on $n$.

This conjecture is a multiplicity refinement of the joints problem, whose corresponding bound for the number of joints is known over arbitrary fields and in every dimension. Without an additional hypothesis, it remains open except when $n=3$ and $k=\mathbf{R}$, a case solved by Iliopoulou.