The universal standard-relations conjecture for MPVs

Let $R_{RDS}$ and $R_{SR}$ be the universal algebras associated respectively with regularized double shuffle relations and standard relations, and let $Z$ denote the map realizing multiple polylogarithm values in the relevant target algebra. For $N=1,2$ take $(R,Z_R)=(\mathbb{R},Z)$, and for a prime $N=p\geq 3$ take $(R,Z_R)=(\mathbb{C},Z)$.

Universal standard-relations conjecture. If $N=1$ or $2$, the map $\varphi_{\mathbb{R}}$ is injective, so the algebra of MPVs is isomorphic to $R_{RDS}$. If $N=p$ is prime with $p\geq 3$, the map $\varphi_{\mathbb{C}}$ is injective, so the algebra of MPVs of level $p$ is isomorphic to $R_{SR}$. Moreover, if $N=3$, then $Z_{LRDS}=Z_{SR}$ and $R_{LRDS}=R_{SR}$. The claim asserts that the listed standard relations are complete in the stated levels; the source gives no resolution.