Lovett's sparse low-rank matrix conjecture

Let $M$ be an $n\times n$ real matrix of rank $r$, and suppose that at most $\varepsilon n^2$ entries of $M$ are nonzero, where $\varepsilon\in(0,1/2)$.

Lovett's conjecture. The matrix $M$ contains an all-zero square submatrix of size at least

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

The conjecture concerns large zero rectangles in sparse low-rank matrices and is motivated by the log-rank problem. The supplied text does not state whether it has been resolved.