Charlton's generalized cyclic insertion conjecture

Let $I^{\mathfrak{m}}(0,a_1,\ldots,a_k,1)$ denote the motivic iterated integral along the specified tangential path from $0$ to $1$. For an odd index $(l_1,\ldots,l_d)$, define $I_{\mathrm{bl}}^{\mathfrak{m}}(l_1,\ldots,l_d)$ by the alternating block construction described in the source. Generalized cyclic insertion conjecture. For $d\geq1$ and $l_1,\ldots,l_d\in\mathbb{Z}_{\geq2}$ such that $\sum_{i=1}^d(l_i-1)$ is odd,

$$\sum_{i=0}^{d-1}I_{\mathrm{bl}}^{\mathfrak{m}}(l_{i+1},\ldots,l_d,l_1,\ldots,l_i)= \begin{cases} I_{\mathrm{bl}}^{\mathfrak{m}}(l_1+\cdots+l_d),&d\text{ is odd},\\ 0,&d\text{ is even}. \end{cases}$$

This motivic block-notation conjecture generalizes the preceding Eulerian and cyclic-insertion conjectures. It is attributed to Charlton, and the source gives no resolution.