The kernel conjecture for formal multiple zeta values

Let $\mathcal{Z}^{\mathrm{f}}_N$ be the weight-$N$ component of the algebra of formal multiple zeta values, and let

$$D_{<N}=\bigoplus_{3\leq 2r+1<N}D_{2r+1},$$

where the maps $D_{2r+1}$ are the derivations defined in the formal multiple zeta value setup.

Kernel conjecture. For every $N\geq 2$,

$$\ker(D_{<N})\cap\mathcal{Z}^{\mathrm{f}}_N=\mathbb{Q}\,\zeta^{\mathrm{f}}(N).$$

This conjecture characterizes the common kernel of the lower-weight derivations in each weight. The source does not state a resolution, so its status is open.