Mirzaei--Suk strengthening of Solymosi's forbidden 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.

Mirzaei--Suk strengthening. For any set of points $P_0$ and any set of lines $\mathcal{L}_0$ in the plane, there is a constant $\varepsilon=\varepsilon(P_0,\mathcal{L}_0)>0$ such that 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 $O(n^{4/3-\varepsilon})$.

This strengthens Solymosi's conjectured little-oh improvement of the Szemerédi--Trotter bound by requiring a fixed power saving depending on the forbidden configuration. The source presents it as open, although related special cases are known.