The Kaneko–Zagier conjecture relating finite and symmetric multiple zeta values

Let $\mathcal{Z}_{\mathcal{A}}$ be the $\mathbb{Q}$-vector subspace of the finite adèlic algebra generated by finite multiple zeta values, and let $\mathcal{Z}/\zeta(2)\mathcal{Z}$ be the quotient of the space of multiple zeta values by the ideal generated by $\zeta(2)$. For an index $\mathds{k}$, write $\zeta_{\mathcal{A}}(\mathds{k})$ for its finite multiple zeta value and $\zeta_S(\mathds{k})$ for its symmetric multiple zeta value.

Kaneko–Zagier conjecture. There exists an isomorphism

$$\mathcal{Z}_{\mathcal{A}} \cong \mathcal{Z}/\zeta(2)\mathcal{Z},\qquad \zeta_{\mathcal{A}}(\mathds{k})\mapsto\zeta_S(\mathds{k}).$$

This is a central conjecture connecting finite and symmetric multiple zeta values; its status is not established by the supplied source.