We mainly answer two open questions about finite multiple harmonic $q$-series on 3-2-1 indices at roots of unity, posed recently by Bachmann, Takeyama, and Tasaka. Two conjectures regarding cyclic sums which generalize the given results are also provided.
"On 3-2-1 values of finite multiple harmonic $q$-series at roots of unity." J. Math. Soc. Japan 74 (3) 753 - 758, July, 2022. https://doi.org/10.2969/jmsj/86238623