On Borodin-Gorin presentation of Heckman-Opdam ${\mathfrak{g}\mathfrak{l}}_n$ hypergeometric function

We prove that Borodin-Gorin presentation for Heckman-Opdam ${\mathfrak{g}\mathfrak{l}}_n$ hypergeometric function by interlacing integrals is equivalent to two other integral presentations studied in arXiv:2508.18864. We also consider related Baxter operators and the corresponding limiting procedure from the Ruijsenaars hyperbolic model.