The kernel conjecture for v-adic multiple zeta values

Let $k$ be the rational function field underlying the paper, let $v$ be a finite place of $k$, and let $\overline{\mathcal{Z}}$ and $\overline{\mathcal{Z}}_v$ denote the corresponding $\overline{k}$-algebras of infinity-adic and $v$-adic multiple zeta values. Theorem~2 gives a surjective $\overline{k}$-algebra homomorphism $\overline{\mathcal{Z}}\twoheadrightarrow\overline{\mathcal{Z}}_v$, and $\zeta_A(q-1)$ is the single zeta value appearing in its known kernel. Kernel conjecture. For every finite place $v$ of $k$, there is an isomorphism

$$\overline{\mathcal{Z}}/(\zeta_A(q-1))\cong\overline{\mathcal{Z}}_v.$$

Equivalently, the kernel of the homomorphism is precisely the principal ideal generated by $\zeta_A(q-1)$. If true, this would make the $v$-adic algebra independent of the finite place up to isomorphism and would imply that it is a graded $\overline{k}$-algebra defined over $k$; the source presents this as open.