GHKK compatibility conjecture for the ${\rm SL}_3$ duality map

Let $\mathbb{I}^{\rm GHKK}$ be the Gross–Hacking–Keel–Kontsevich duality map for the cluster varieties associated with the mutation class of the $3$-triangulation quiver, and let $\mathbb{I}^{\rm br}$ and $\mathbb{I}$ be the geometric duality maps constructed in the paper. GHKK compatibility conjecture. The maps $\mathbb{I}^{\rm GHKK}$ and $\mathbb{I}^{\rm br}$ coincide, or equivalently $\mathbb{I}^{\rm GHKK}$ coincides with $\mathbb{I}$. The comparison concerns whether the abstract GHKK construction agrees with the paper's geometric construction; the source gives no resolution.