Regular μ-independence of Carlitz zeta values and the Carlitz period

Let $A=\mathbb{F}_q[\theta]$, let $p$ be the characteristic of $\mathbb{F}_q$, and let $K_s$ and the notion of regular $\mu$-independence be as in the source. Consider the set consisting of $\widetilde{\pi}$ and the values $\zeta_A(n;s)$ satisfying

$$p\nmid n,\qquad n\not\equiv s\pmod{q-1}.$$

Regular-independence conjecture. Every finite subset of this set is regularly $\mu$-independent over $K_s$.

The source states that this does not seem to follow from the preceding functional conjecture and is probably difficult. No resolution is supplied.