  The Laplacian of a (weighted) Cayley graph on the Weyl group $W(B_n)$ is a
$N\times N$ matrix with $N = 2^n n!$ equal to the order of the group. We show
that for a class of (weighted) generating sets, its spectral gap (lowest
nontrivial eigenvalue), is actually equal to the spectral gap of a $2n \times
2n$ matrix associated to a $2n$-dimensional permutation representation of
$W_n$. This result can be viewed as an extension to $W(B_n)$ of an analogous
result valid for the symmetric group, known as `Aldous' spectral gap
conjecture', proven in 2010 by Caputo, Liggett and Richthammer.
