I believe this exact result is stated as Corollary 6.8 in this preprint of Soichi Okada: https://arxiv.org/abs/1904.03386v1. Okada attributes the result to A.O. Morris with citation: <cite authors="Morris, A. O.">_Morris, A. O._, [**A note on the multiplication of Hall functions**](http://dx.doi.org/10.1112/jlms/s1-39.1.481), J. Lond. Math. Soc. 39, 481-488 (1964). [ZBL0125.01702](https://zbmath.org/?q=an:0125.01702).</cite>

EDIT: Ah, sorry, this is a reference for the "dual" result, which you mentioned you already know about.