The geometric interpretation conjecture for cluster-monomial MV-polynomials

Let $\mathbf p$ be a Kostant picture such that the MV-polynomial $\mathcal P_{\mathbf p}$ is a cluster monomial. Let $M=M(\widetilde{\mathbf p})$ be the corresponding MV-cycle with lowest coweight $\mu$, and let $M^\circ$ be the dense subset of $M\cap(\mathsf N(\mathbb C[t,t^{-1}])\cdot\underline\mu)$. Define $\chi_1$ from the coefficient of $t^d$ in $\mathcal P_{\mathbf p}(\mathbf n)$, and define $\chi_2$ as the Jacobian of the induced finite-dimensional projection map between the spaces associated with the left and right ends of the loops of $\mathbf p$.

The geometric interpretation conjecture. The function $\chi_1$ is well-defined and

$$\chi_1,\chi_2:M^\circ\longrightarrow\mathbb C$$

are identical.

This conjecture identifies the cluster-monomial MV-polynomial coefficient with a geometric Jacobian on the corresponding MV-cycle, linking the algebraic and geometric descriptions of MV data.