An integral with respect to the Haar measure on a unitary group

Let $A={rm diag}[a_1,dots,a_n]$ and $B={rm diag}[b_1,dots,b_n]$. Let $Delta(a)$ be the Vandermonde product in the $a_j$, and similarly $Delta(b)$ be the Vandermonde product in the $b_k$. Suppose ${rm min} , a_j ge {rm max} , b_k$. Let ${rm d}U$ denote the normalised Haar measure on the unitary group $U(n)$. Then by Eq. (3.21) in Gross and Richards, "Total positivity, spherical series, and hypergeometric functions of matrix argument", J. Approx. Th. vol. 59 (1989): 224-246, $$ int_U det(A - U B U^dagger)^p {rm d}U = c_{n,p} {det [ (a_j - b_k)^{p+n-1}]_{j,k=1}^n over Delta(a) Delta(b)}, $$ where $c_{n,p} = prod_{j=0}^{n-1} binom{p+n-1}{j}^{-1}$. Setting $p=1$, subject to the condition ${rm min} , a_j ge {rm max} , b_k$, this is the sought evaluation. See Theorem 2.3 in Kieburg, Kuijlaars and Stivigny, "Singular value statistics of matrix products with truncated unitary matrices", IMRN vol. 2016 (2016) 3392-3424 for a generalisation.