The extended double-shuffle completeness conjecture for Euler sums

Let $R$ be a commutative $\mathbb{Q}$-algebra with $1$, and let $Z_R$ be a map from the admissible word algebra $\mathcal{A}^0$ to $R$ satisfying the finite double-shuffle property. Let $R_{\mathrm{EDS}}$ be the universal algebra for maps satisfying the extended double-shuffle property, with universal map $Z_{\mathrm{EDS}}:\mathcal{A}^0\to R_{\mathrm{EDS}}$ and induced map $\varphi_R:R_{\mathrm{EDS}}\to R$.

Extended double-shuffle conjecture. The induced map $\varphi_R$ is injective; equivalently, the algebra of Euler sums is isomorphic to $R_{\mathrm{EDS}}$.

This conjecture asserts that the extended double-shuffle relations give a complete presentation of the algebra of Euler sums. The source supplies no evidence of a resolution.