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: Morris, A. O., A note on the multiplication of Hall functions, J. Lond. Math. Soc. 39, 481-488 (1964). ZBL0125.01702.
EDIT: Ah, sorry, this is a reference for the "dual" result, which you mentioned you already know about.