Conjecture on the correspondence between Tate-algebra zeta values and periodic multiple sums

Let $A$ be the coefficient ring and let $\bigsqcup$ denote disjoint union of composition arrays. For each composition array $\mathcal{C}$, write $\zeta_A(\mathcal{C})$ for the associated multiple zeta value in a Tate algebra and $\varphi_A(\mathcal{C})$ for the associated $A$-periodic multiple sum. Let $\mathcal{Z}_\zeta$ and $\mathcal{Z}_\varphi$ be the corresponding $\mathbb{F}_p$-algebras.

The correspondence conjecture. The correspondence $\zeta_A(\mathcal{C})\leftrightarrow\varphi_A(\mathcal{C})$ induces an isomorphism of $\mathbb{F}_p$-algebras

$$\mathcal{Z}_\zeta\cong\mathcal{Z}_\varphi.$$

The claim would identify the algebra of multiple zeta values in Tate algebras with the algebra of $A$-periodic multiple sums. The source gives no resolution, so the conjecture remains open.