Zero distribution conjecture for $v$-adic multiple zeta values over $\mathbb{F}_q(t)$

Let $K=\mathbb{F}_q(t)$, let $v$ be any monic prime, and let $\mathbf{s}=(s_1,\ldots,s_r)\in S_v^r$, where $S_v$ is the space of $v$-adic weights defined by

$$S_v:=\lim\limits_{\substack{\longleftarrow\\ n}}\mathbb{Z}/(q^d-1)p^n\mathbb{Z}$$

for $d=\deg v$. An element $s=c_{-1}+\sum_{i=0}^{\infty}c_i(q^d-1)p^i\in S_v$ is $q$-even when $q-1\mid c_{-1}$. A multiple zeta value is called a trivial zero when it vanishes for the trivial-zero condition established for depth greater than one.

Zero distribution conjecture. The $v$-adic multiple zeta value satisfies

$$\zeta_v(\mathbf{s})=0$$

if and only if one of the following conditions holds:

1. $r>1$ and $\mathbf{s}$ is a trivial zero;
2. $r=1$ and $s$ is $q$-even.

For degree-one primes this is proved in the preceding theorem; the conjecture extends the same characterization to arbitrary monic primes $v$.