  In order to give a unified generalization of the BW inequality and the DDVV
inequality, Lu and Wenzel proposed three Conjectures 1, 2, 3 and an open
Question 1 in 2016. In this paper we discuss further these conjectures and put
forward several new conjectures which will be shown equivalent to Conjecture 2.
In particular, we prove Conjecture 2 and hence all conjectures in some special
cases. For Conjecture 3, we obtain a bigger upper bound $2+\sqrt{10}/2$, and we
also give a weaker answer for the more general Question 1. In addition, we
obtain some new simple proofs of the complex BW inequality and the condition
for equality.
