Double-shuffle kernel conjecture for refined symmetric multiple zeta values

Let $\mathfrak{h}=\bigoplus_{k=-1}^{\infty}\mathfrak{h}_{k}$ be graded by

$$\mathfrak{h}_{k}=\bigoplus_{a_{1},\dots,a_{k+1}\in\{0,1\}}\mathbb{Q}e_{a_{1}}\cdots e_{a_{k+1}},$$

and write $\mathfrak{h}_{k}^{0}=\mathfrak{h}_{k}\cap\mathfrak{h}^{0}$. For $u,v,w\in\mathfrak{h}^{0}$, let

$$D(u,v,w):=u*(v\mathbin{\shuffle}w)-v\mathbin{\shuffle}(u*w),$$

where $Z^{RS}$ is the refined symmetric multiple-zeta-value map. Double-shuffle kernel conjecture. For every $k\in\mathbb{Z}$, the space $\ker Z^{RS}\cap\mathfrak{h}_{k}$ is spanned by

$$\left\langle D(u,v,w)\mid a+b+c=k-1,\ (u,v,w)\in\mathfrak{h}_{a}\times\mathfrak{h}_{b}\times\mathfrak{h}_{c}\right\rangle.$$

This conjecture asserts that all relations in the kernel of the refined symmetric multiple-zeta-value map arise from the displayed double-shuffle expressions. The source gives no resolution status.