Lovett's incidence-graph conjecture for points and hyperplanes

Let $\varepsilon\in(0,1/2)$, and consider the incidence graph between $n$ points and $n$ hyperplanes in $\mathbb{R}^d$, with an edge joining a point to a hyperplane when the point lies on that hyperplane.

Lovett's incidence-graph conjecture. If the graph has at least $(1-\varepsilon)n^2$ edges, then it contains a complete balanced bipartite subgraph of size at least

$$n\cdot\exp(-O(\sqrt{\varepsilon d})).$$

This is presented as an equivalent geometric formulation of Lovett's sparse low-rank matrix conjecture. The supplied text does not state whether it has been resolved.