An analogue of Hirose's relation for finite multiple harmonic $q$-series at roots of unity

Recently, Hirose proved an analogue of the linear part of Kawashima's relations for refined symmetric multiple zeta values. In this paper, we establish an analogue of Hirose's relation for finite multiple harmonic $q$-series at roots of unity. As an application, we prove the cyclic sum conjecture proposed by Kh. Hessami Pilehrood, T. Hessami Pilehrood, and R. Tauraso.