The class-number-one zetalike multizeta-value conjecture

Let $A$ be a class-number-one coefficient ring with constant field $\mathbb{F}_q$. For positive integers $n$ and $k$, let $\zeta(s_1,\ldots,s_r)$ denote the multizeta value associated with positive integer arguments, and call it zetalike when

$$\frac{\zeta(s_1,\ldots,s_r)}{\zeta(s_1+\cdots+s_r)}\in K.$$

The class-number-one zetalike conjecture. The multizeta values

$$\zeta\bigl(q^n-1,(q-1)q^n,\ldots,(q-1)q^{n+k}\bigr)$$

are zetalike.

In the genus-zero case $A=\mathbb{F}_q[t]$, the corresponding explicit identity was conjectured and is now proved in all depths; in higher genus, this family is the one identified by the authors' exploration, with the conjecture remaining open in the stated generality.