A $q$-recurrence for a finite Apéry limit

The Kaneko-Zagier conjecture predicts a correspondence between finite and symmetric multiple zeta values. Under this correspondence, $\zeta (3)$ corresponds to an element $Z(3)$ defined by Bernoulli numbers. We prove a conjecture of Tasaka relating $Z(3)$ to the quotient of two solutions of a recurrence. A two-index $q$-recurrence connects this quotient to a finite harmonic $q$-series. Using a method of the author, Takeyama, and Tasaka, we obtain the algebraic and analytic limits $3Z(3)\mathrm{/}4$ and $3\zeta (3)\mathrm{/}4$ at roots of unity.