The constancy conjecture for the regulator on the Lie coalgebra

Let $\mathbb L(F)$ be the Lie coalgebra of multiple polylogarithms over a field $F$, with cobracket $\delta$, and let $\mathbb L^{\operatorname{symb}}(\mathbb C)$ be its symbolic version. Consider the map

$$r\colon\mathbb L^{\operatorname{symb}}(\mathbb C)\to\mathbb R,$$

which sends $[x_1,\dots,x_d]_{n_1,\dots,n_d}$ to $\mathcal L_{n_1,\dots,n_d}(x_1,\dots,x_d)$ and $[x]_0$ to $\log(|x|)$. Constancy conjecture. If $\alpha\in\mathbb L(\mathbb C(t))$ satisfies $\delta(\alpha)=0$, then $r(\alpha(t))=0$ is constant in $t$. This predicts that elements in the kernel of the cobracket give constant real-valued regulators under specialization, generalizing the preceding result for the corresponding regulator map. The source provides no resolution status for this conjecture.