CMZV membership conjecture for polylogarithms at admissible values

Let $N$ be a positive integer and let

$$\mathcal{C}^N=\{R(0)\mid R(x)\text{ is }N\text{-admissible}\}.$$

For a word $w\in\{x_0,x_1\}^\ast$ of weight $n$, let $\operatorname{Li}_w(x)$ denote the corresponding multiple polylogarithm, and let $\textsf{CMZV}^N_w$ denote the space of cyclotomic multiple zeta values of level $N$ and weight $n$ associated with $w$. CMZV membership conjecture. For every $x\in\mathcal{C}^N$,

$$\operatorname{Li}_w(x)\in\textsf{CMZV}^N_w.$$

The source notes that the sets of $N$-admissible rational functions are not known with certainty and gives conjectural descriptions of $\mathcal{C}^2$ and $\mathcal{C}^4$; the asserted membership of these polylogarithmic values in the corresponding CMZV spaces remains open.