Ihara–Kaneko–Zagier conjecture on algebraic relations among multiple zeta values

Let $\mathcal{Z}$ be the algebra of multiple zeta values, and let $\mathbb{Q}\langle X\rangle$ and $\mathbb{Q}\langle Y\rangle$ be the non-commutative free algebras on $X=\{x_0,x_1\}$ and $Y=\{y_1,y_2,\ldots\}$, respectively, equipped with the shuffle product $\shuffle$ and stuffle product $\ast$. The extended maps to $\mathcal{Z}$ send $x_0$ and $x_1$ to $0$, and $y_1$ to $0$.

Ihara–Kaneko–Zagier conjecture. All algebraic relations in the algebra $\mathcal{Z}$ are obtained from the comparison of the extended maps

$$(\mathbb{Q}\langle X\rangle,\shuffle)\to\mathcal{Z}$$

and

$$(\mathbb{Q}\langle Y\rangle,\ast)\to\mathcal{Z}$$

via the regularization map from the cited theorem.

This is a central open problem concerning whether shuffle and stuffle relations, together with regularization, generate all algebraic relations among multiple zeta values.