Harmonic-product conjecture for the algebra of Frobenius-twisted zeta values

Let $\mathbb{I}$ be the inversive $\mathbb{F}_p$-algebra generated by $\mu^m(\zeta_A(1,\chi_{t_i}))$ for $i\in\mathbb{N}^*$ and $m\in\mathbb{Z}$, where $\mu(c)=c^p$. The harmonic product gives algebraic relations among these generators.

Harmonic-product conjecture. The only $\mathbb{F}_p$-algebraic relations in $\mathbb{I}$ are those coming from the harmonic product.

This is a transcendence-style independence assertion for the Frobenius-twisted Tate-algebra zeta values. The source gives no resolution, so it remains open.