Supercongruences of multiple harmonic $q$-sums and generalized finite/symmetric multiple zeta values

The Kaneko–Zagier conjecture describes a correspondence between finite multiple zeta values and symmetric multiple zeta values. Its refined version has been established by Jarossay, Rosen and Ono–Seki–Yamamoto. In this paper, we explicate these conjectures through studies of multiple harmonic $q$-sums. We show that the (generalized) finite/symmetric multiple zeta value are obtained by taking an algebraic/analytic limit of multiple harmonic $q$-sums. As applications, new proofs of reversal, duality and cyclic sum formulas for the generalized finite/symmetric multiple zeta values are given.