Gezmis–Pellarin injectivity conjecture for the map \mathcal{G}_{\Sigma}

Let $K$ be the field and let $\Sigma$ be the index set used to define the spaces of multiple polylogarithms and multiple zeta values. Let

$$\mathcal{G}_{\Sigma}\colon \mathcal{Z}^{\text{triv}}_{\Sigma}(K)\longrightarrow \mathcal{Z}_{\emptyset}(K)$$

be the map obtained by composing the map from $\mathcal{Z}^{\text{triv}}_{\Sigma}(K)$ to $\mathcal{L}_{\Sigma}(K)$ with evaluation at $X_i=1$ for all $i\in\Sigma$. Gezmis–Pellarin's injectivity conjecture. The map $\mathcal{G}_{\Sigma}$ is injective. This conjecture concerns the relations among Thakur's multiple zeta values arising from the two combinatorial constructions of multiple polylogarithms. Its resolution is not specified in the source text.