Solymosi's forbidden point-line configuration conjecture

Let $P$ be a set of $m$ points and let $\mathcal{L}$ be a set of $n$ lines in the plane. Write $I(P,\mathcal{L})$ for the set of incidences between points of $P$ and lines of $\mathcal{L}$, and let $G(P,\mathcal{L})$ be their incidence graph. Two configurations $(P_1,\mathcal{L}_1)$ and $(P_2,\mathcal{L}_2)$ are isomorphic when their incidence graphs are isomorphic.

Solymosi's conjecture. For any set of points $P_0$ and any set of lines $\mathcal{L}_0$ in the plane, the maximum number of incidences between $n$ points and $n$ lines in the plane containing no subconfiguration isomorphic to $(P_0,\mathcal{L}_0)$ is $o(n^{4/3})$.

The conjecture asks whether forbidding every fixed point-line configuration improves the sharp Szemerédi--Trotter bound. It is known in several special cases, including configurations arising from point sets in general position and point sets that do not contain grids, but remains open in general.