  Mutually unbiased bases correspond to highly useful pairs of measurements in
quantum information theory. In the smallest composite dimension, six, it is
known that between three and seven mutually unbiased bases exist, with a
decades-old conjecture, known as Zauner's conjecture, stating that there exist
at most three. Here we tackle Zauner's conjecture numerically through the
construction of Bell inequalities for every pair of integers $n,d \ge 2$ that
can be maximally violated in dimension $d$ if and only if $n$ MUBs exist in
that dimension. Hence we turn Zauner's conjecture into an optimisation problem,
which we address by means of three numerical methods: see-saw optimisation,
non-linear semidefinite programming and Monte Carlo techniques. All three
methods correctly identify the known cases in low dimensions and all suggest
that there do not exist four mutually unbiased bases in dimension six, with all
finding the same bases that numerically optimise the corresponding Bell
inequality. Moreover, these numerical optimisers appear to coincide with the
"four most distant bases" in dimension six, found through numerically
optimising a distance measure in [P. Raynal, X. L\"u, B.-G. Englert, Phys. Rev.
A, 83 062303 (2011)]. Finally, the Monte Carlo results suggest that at most
three MUBs exist in dimension ten.
