The determinant conjecture for cluster monomials and MV-polynomials

Let $\mathsf G=\operatorname{GL}_n(\mathbb C)$, let $\mathsf N$ be the unipotent radical of a Borel subgroup, and let $\mathbb C[\mathsf N]$ be equipped with its Fomin–Zelevinsky cluster algebra structure. For a Kostant picture $\mathbf p$, let $\mathcal P_{\mathbf p}$ be the corresponding MV-polynomial, and let $A_{\mathbf p}$ be the associated matrix; write $\hat A_{\mathbf p}$ for a matrix obtained from $A_{\mathbf p}$ by setting some entries to zero.

The determinant conjecture. Every cluster monomial is an MV-polynomial that is naturally expressible as a determinant: if $\psi$ is a cluster monomial, then there is a Kostant picture $\mathbf p$ such that

$$\psi=\mathcal P_{\mathbf p}=\operatorname{det}(\hat A_{\mathbf p}).$$

This predicts a direct relation between cluster algebra combinatorics and the MV-polynomial basis of $\mathbb C[\mathsf N]$, while specifying a broad class of MV-polynomials with determinant formulas. The source presents this as a conjecture and gives examples, including cases where a prescribed determinant construction fails outside the stated class.