Generalized Kaneko–Zagier homomorphism conjecture for colored multiple zeta values

Let $N$ be a positive integer and let $\alpha\in(\mathbb{Z}/N\mathbb{Z})^\times$. Let $\mathcal{Z}^{\mathcal{S}(N;\alpha)}$ and $\mathcal{Z}^{\mathcal{A}(N;\alpha)}$ be the $\mathbb{Q}(\zeta_N)$-algebras spanned by symmetric and finite colored multiple zeta values of level $N$ and class $\alpha$. For their indexed generators, write $L^{\mathcal{S}}_\alpha\binom{\boldsymbol{\eta}}{\boldsymbol{k}}$ and $L^{\mathcal{A}}_\alpha\binom{\boldsymbol{\eta}}{\boldsymbol{k}}$.

Generalized Kaneko–Zagier conjecture. For each $\alpha\in(\mathbb{Z}/N\mathbb{Z})^\times$, the $\mathbb{Q}(\zeta_N)$-linear map

$$\mathcal{Z}^{\mathcal{S}(N;\alpha)}\longrightarrow\mathcal{Z}^{\mathcal{A}(N;\alpha)},\qquad L^{\mathcal{S}}_\alpha\binom{\boldsymbol{\eta}}{\boldsymbol{k}}\longmapsto L^{\mathcal{A}}_\alpha\binom{\boldsymbol{\eta}}{\boldsymbol{k}}$$

is a well-defined algebra homomorphism whose kernel is generated by $2\pi i$. The preceding relations support this conjecture, but the source gives no resolution.