Vanishing conjecture for Carlitz finite zeta values

Let $n>0$ and $s\geq 0$ be integers. Write the base-$q$ expansion of $n$ as $n=n_0+n_1q+\cdots+n_rq^r$ and set $\ell_q(n):=\sum_i n_i$. For the finite zeta value of level $s$ and exponent $n$, $Z_{\mathcal{A}}(n;s)\in\mathcal{A}_s$, one has the following properties. Vanishing conjecture. If $n\not\equiv s\pmod{q-1}$, then

$$Z_{\mathcal{A}}(n;s)\neq 0.$$

If $n\equiv s\pmod{q-1}$, then

$$Z_{\mathcal{A}}(n;s)=0\quad\Longleftrightarrow\quad \ell_q(n)>s.$$

This predicts exactly when these Carlitzian finite zeta values vanish, while asserting nonvanishing in the complementary congruence class. The source provides no resolution status for the conjecture.