Motivic variation conjecture for the Grassmannian n-logarithm

Let $\widehat G_{n-1}^n$ be the configuration space used to define the Grassmannian $n$-logarithm, and let $a_i,b_j$ be the maps on this space appearing in its functional equations. A framed mixed Tate motive has a Lie period given by the Lie period of its Hodge realization. Motivic variation conjecture. There exists a variation $\mathbb L_n^{\mathcal M}$ of framed mixed Tate motives over $\widehat G_{n-1}^n$ such that

$$\sum_{i=0}^{2n}(-1)^i a_i^*\mathbb L_n^{\mathcal M}=0,\qquad \sum_{j=0}^{2n}(-1)^j b_j^*\mathbb L_n^{\mathcal M}=0,$$

and whose Lie period $\mathcal L_n^{\mathcal M}$ satisfies

$$\mathcal L_n^{\mathcal M}-\mathcal L_n^G=\sum_{i=0}^{2n-1}(-1)^i a_i^*F_n,$$

where $F_n$ is a function on $\widehat G_{n-2}^n(\mathbb C)$. Moreover, the functional equations for $\mathcal L_n^{\mathcal M}$ essentially determine it: the space of smooth or measurable functions satisfying them is finite dimensional. This conjecture seeks a motivic interpretation and finite-dimensionality statement for the Grassmannian $n$-logarithm; the source gives no resolution.