The monochromatic-rectangle formulation of the log-rank conjecture

Let $M$ be a Boolean matrix, and let a monochromatic rectangle be a submatrix all of whose entries have the same value. Let ${\mathrm{r}}(M)$ denote the rank of $M$ over the reals, and define the density of a rectangle as its number of entries divided by the total number of entries of $M$. Monochromatic-rectangle formulation. Any Boolean matrix $M$ has a monochromatic rectangle of density

$$2^{-\operatorname{polylog}({\mathrm{r}}(M))}.$$

The source identifies this as an equivalent version of the log-rank conjecture due to Nisan and Wigderson. Since the log-rank conjecture is stated to remain unsolved, this formulation is also open.