Effect of protocol ordering on multiparty comm. complexity
Any protocol $\pi$ can be modified into an equivalent protocol $\hat\pi$ that has the special "round-robin" communication pattern. The modification is as follows: Whenever party $i$ generates an output in $\pi$, it holds it in a buffer until party $i-1$ has spoken. After party $i-1$ speaks, party $i$ either releases its buffer or broadcasts a "dummy" (or empty) message if there is nothing in the buffer.

The conversion of $\pi$ into $\hat\pi$ is without loss of generality, with respect to the correctness and security properties of the protocol. Whether it incurs a loss of generality with respect to communication complexity depends on whether the model allows "empty" messages. You should see whether the proof technique in this paper assumes that parties can broadcast only non-empty messages. If empty messages are allowed, then $\hat\pi$ has the same communication complexity as $\pi$.