Pellarin's expansion conjecture for Tate-algebra zeta values

Let $q=p^e$ with $e>0$, let $\mu$ be the $\mathbb{F}_p$-linear Frobenius automorphism $c\mapsto c^p$, and let $\mathbb{I}$ be the inversive $\mathbb{F}_p$-difference algebra generated by all $\mu^m(\zeta_A(1,\chi_{t_i}))$ for $i\in\mathbb{N}^*$ and $m\in\mathbb{Z}$. Set $\zeta_A(0)=1$.

Pellarin's expansion conjecture. For all $n\in\mathbb{N}^*$ and $\Sigma\subset\mathbb{N}^*$ satisfying $|\Sigma|\equiv n\pmod{q-1}$, there is a unique expansion

$$\zeta_A(n;\sigma_\Sigma)=\sum_{\substack{0\leq k\leq n\\ k\equiv0\pmod{q-1}}}\zeta_A(k)\eta_k,\qquad \eta_k\in\mathbb{I}.$$

This predicts that every indicated Tate-algebra zeta value has a unique expansion over the algebra generated by Frobenius twists of weight-one zeta values. The source presents examples and related identities but no resolution, so the conjecture remains open.