  In this note, we study the asymptotics of a spherical integral that is a
multiplicative counterpart to the well-known Harish-Chandra Itzykson Zuber
integral. This counterpart can also be expressed in terms the Heckman-Opdam
hypergeometric function. When the argument of this spherical integral is of
finite support and of order $N$, these asymptotics involve a modified version
of the $S$-transform of the limit measure of the matrix argument and its
largest eigenvalue. To prove our main result, we are leveraging a technique of
successive conditionning. In particular we prove in a "mathematically rigorous"
manner a result from Mergny and Potters in the case $\beta =1,2$ and we
generalize it for multiple arguments
