Probabilistic Communication Complexity of Boolean Relations

In [KW] it was proved that for every boolean function $f$ there exist a communication complexity game $R_f$ such that the minimal circuit-depth of $f$ exactly equals to the communication complexity of $R_f$. If $f$ is monotone then there also exists a game $R_f^m$ with communication complexity exactly equals to the monotone depth of $f$. It was also proved in [KW] that the communication complexity of $R_{st-connectivity}^m$ is $\Omega(log^2 n)$, or equivalently that the monotone depth of the $st$-connectivity functions is $\Omega(log^2 n)$.