The allowable-path determinant conjecture for acyclic Kostant pictures

Let $\mathbf p$ be a Kostant picture whose graph $G_{\mathbf p}$ is acyclic, and order its loops. Let $A_{\mathbf p}$ be the associated matrix. An allowable path in $G_{\mathbf p}$ is a path of distinct vertices satisfying the two orientation and encircling conditions specified in the source. Define $\hat A_{\mathbf p}$ by changing its $ij$-th entry to $0$ whenever there is no allowable path from the $j$-th loop to the $i$-th loop.

The allowable-path determinant conjecture. For every such Kostant picture,

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

This is a concrete determinant formula for MV-polynomials associated with acyclic Kostant pictures. The paper reports that it holds for many additional examples, but that the displayed graph construction does not cover some cyclic examples and may not be the natural general condition.