Bukh–Matoušek–Pálvölgyi k-fan incidence conjecture

Fix an integer $k\geq 3$. Let $P$ be a set of $n$ points and let $\mathcal{L}$ be a set of $n$ lines in the plane. A $k$-fan is a configuration consisting of distinct points $p_0,p_1,\ldots,p_k\in P$ and distinct lines $l_0,l_1,\ldots,l_k\in\mathcal{L}$ such that

$$p_0\in l_1\cap l_2\cap\cdots\cap l_k,\qquad p_i\in l_0\cap l_i\quad\text{for }i=1,\ldots,k,$$

and $p_0\notin l_0$. Bukh–Matoušek–Pálvölgyi's $k$-fan incidence conjecture. If $(P,\mathcal{L})$ does not contain a $k$-fan, then the number of incidences satisfies

$$|I(P,\mathcal{L})|=o(n^{4/3}).$$

The case $k=2$ is known by work of Solymosi, whereas the conjecture for every fixed $k\geq 3$ remains open.