Conjecture on the commutation relations of the geometric operators

Let $(n,k)$ and $(n',k')$ be lattice points, and let $e_{n,k}$ and $e_{n',k'}$ be the corresponding geometric operators. Let $\Delta_*$ denote pushforward along the diagonal, let $\phi:\BK\to K_S$ be the map from the constants, and let the sum range over convex paths $v$ with the same indexing and coefficients $p^{n,n_1,\ldots,n_t,n'}_{k,k_1,\ldots,k_t,k'}(q_1,q_2)$ as in the shuffle-algebra commutation relation. The geometric commutation conjecture. For any lattice points $(n,k)$ and $(n',k')$,

$$[e_{n,k},e_{n',k'}]=\Delta_*\left(\sum_{v\ \emph{convex path}}\phi\left(p^{n,n_1,\ldots,n_t,n'}_{k,k_1,\ldots,k_t,k'}(q_1,q_2)\right)\cdot e_{n_1,k_1}\cdots e_{n_t,k_t}\Big|_\Delta\right).$$

This is the geometric analogue of the defining commutation relations of the double shuffle algebra and is intended to identify the geometric operators with its generators. The source states the relation as a conjectural implication and does not provide a proof or resolution.