As far as I know, the factorization norm lower bound given by Linial and Shraibman is essentially the only lower bound known for quantum communication complexity (or at least it subsumes all others). Is there any evidence against this bound being tight?
The factorization norm bound (also called the $\gamma_2$ bound) I speak of is Theorem 13 of Linial, Shraibman 2008. In fact, this bound follows from a reduction from quantum communication complexity to the bias in a 2-player XOR game Degorre, et al. 2008. For this reason it could be expected to be a lousy bound since the XOR game doesn't even have anything to do with communication. For the impatient, a brief overview is given in some slides by Troy Lee.
The introduction text of Jain, Klauck 2010 says that information theoretic techniques may offer some competition but it is not known whether these beat the $\gamma_2$ bound. So it would seem that, at least as of a few years ago, $\gamma_2$ was the best technique. But I would like to know if there is even a specific example of a function that is believed to have quantum communication complexity much greater than the $\gamma_2$ bound.