  Recently, we performed a two $\beta$-ensemble realization of the series of
$\beta$-deformed WLZZ matrix models involving $\beta$-deformed
Harish-Chandra-Itzykson-Zuber integrals. The realization was derived and
studied by using Ward identities, which do not allow one to fix integration
contours, these latter were chosen to be real axis for one $\beta$-ensemble and
imaginary axis for the other one basing on some particular checks. Here, we
evaluate the $\beta$-ensemble integrals directly using a conjecture by I.G.
Macdonald, and explain that another choice of integration contours is also
possible.
