$\beta$-WLZZ models from $\beta$-ensemble integrals directly

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 based 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.