The depth-strengthening conjecture for block-to-depth relations

Let $\xi_B$ and $\xi_D$ be the block-graded and depth-graded functionals associated with the relations in the preceding block-to-depth reduction theorem, and let $\sigma$ be an element of the Lie ideal generated by the exceptional generators in ${\mathfrak{d}}{\mathfrak{g}}$ of depth $r-1$. Depth-strengthening conjecture. The relations from the preceding theorem hold modulo terms of lower depth; explicitly,

$$\langle\xi_B-2^{r-1}\xi_D,\sigma\rangle=0.$$

The source reports numerical evidence for this strengthening and relates it to Brown's unevenness conjecture and the homological Broadhurst–Kreimer conjecture; it remains open in general.