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

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