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

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