The refined Kaneko–Zagier conjecture for completed finite and symmetric multiple zeta values

Let $\boldsymbol{k}$ be an index, let $\operatorname{wt}(\boldsymbol{k})$ denote its weight, and let $\zeta_{\widehat{\mathcal{A}}}(\boldsymbol{k})$ and $\zeta_{\widehat{\mathcal{S}}}(\boldsymbol{k})$ denote the completed finite and completed symmetric multiple zeta values, respectively. Let $\boldsymbol{p}$ be the distinguished element of $\widehat{\mathcal{A}}$, and let $t$ be a formal variable. For a positive integer $k$ and rational numbers $c_{\boldsymbol{k}}^{(l)}$ indexed by $l\ge0$ and indices of weight $k+l$, A refined version of the Kaneko–Zagier conjecture.

$$\sum_{l\ge0}\boldsymbol{p}^{l}\left(\sum_{\operatorname{wt}(\boldsymbol{k})=k+l}c_{\boldsymbol{k}}^{(l)}\zeta_{\widehat{\mathcal{A}}}(\boldsymbol{k})\right)=0\quad\text{in }\widehat{\mathcal{A}}$$

if and only if

$$\sum_{l\ge0}t^{l}\left(\sum_{\operatorname{wt}(\boldsymbol{k})=k+l}c_{\boldsymbol{k}}^{(l)}\zeta_{\widehat{\mathcal{S}}}(\boldsymbol{k})\right)=0\quad\text{in }\overline{\mathcal{Z}}[[t]].$$

This refines the correspondence between relations among finite and symmetric multiple zeta values by retaining the weight filtration through powers of $\boldsymbol{p}$ and $t$. The supplied text describes it as conjectural and gives no resolution status.