  Excluding the existence of four MUBs in $\bbC^6$ is an open problem in
quantum information. We investigate the number of product vectors in the set of
four mutually unbiased bases (MUBs) in dimension six, by assuming that the set
exists and contains a product-vector basis. We show that in most cases the
number of product vectors in each of the remaining three MUBs is at most two.
We further construct the exceptional case in which the three MUBs respectively
contain at most three, two and two product vectors. We also investigate the
number of vectors mutually unbiased to an orthonormal basis.
