Large zero rectangle conjecture for low-rank sparse matrices

Let $M$ be an $n \times n$ real matrix with $\operatorname{rank}(M)=r$ and such that $M_{i,j} \ne 0$ for at most $\varepsilon n^2$ entries. A large zero rectangle conjecture asserts that there exist $A,B \subset [n]$ such that

$$M_{a,b}=0 \qquad \forall a \in A, b \in B$$

and

$$|A|,|B| \ge n \cdot \exp(-O(\sqrt{\varepsilon r})).$$

This conjecture seeks to generalize the large monochromatic-rectangle phenomenon used in deterministic communication complexity from Boolean matrices to low-rank sparse real matrices. The supplied text gives no evidence that the conjecture has been resolved.