A note on the sum of finite multiple harmonic $q$-series on $rtext{-}(r+1)$ indices

We study the sum of the finite multiple harmonic $q$-series on $rtext{-}(r+1)$ indices at roots of unity with $r=1,2,3$. And we give the equivalent conditions of two conjectures regarding cyclic sums of finite multiple harmonic $q$-series on $1text{-}2text{-}3$ indices at roots of unity, posed recently by Kh. Pilehrood, T. Pilehrood and R. Tauraso.