Why does the log-rank conjecture use rank over the reals? In communication complexity, the log-rank conjecture states that

$$cc(M) = (\log rk(M))^{O(1)}$$

Where $cc(M)$ is the communication complexity of $M(x,y)$ and $rk(M)$ is the rank of $M$ (as a matrix) over the reals.

However, when you are just using the rank-method to lower bound $cc(M)$ you can use $rk$ over any field that is convenient. Why does the log-rank conjecture restrict to rk over the reals? Is the conjecture resolved for $rk$ over fields of non-zero characteristic?  If not, is it of interest or is there something special about $rk$ over $\mathbb{R}$?
The conjecture fails over $\mathbb{F}_2$. Look at $M(x, y) = \langle x, y \rangle \bmod 2$, and $x, y \in \{0, 1\}^n$. The communication complexity is $\Omega(n)$, but the rank of $M$ over $\mathbb{F}_2$ is $n$, by the linearity of inner product.