Resolved noncrossing formula for the biadjoint scalar amplitude

Let $u_1,\ldots,u_n\in\mathbb{C}^3$ be the columns of a matrix, let $\mathbf{NC}_{3,n}$ be the noncrossing complex of nonfrozen $3$-subsets, and let $\hat{m}^{(3),NC}_n(\mathbb{I},\mathbb{I})$ denote the resolved biadjoint scalar amplitude. The resolved minors and potential are defined by the formulas in the source, and $\hat{\eta}_J(s)$ are linear functions of the kinematic variables.

Resolved amplitude conjecture. There exists a set of linear functions $\hat{\eta}_J(s)$ such that

$$\hat{m}^{(3),NC}_n(\mathbb{I},\mathbb{I})=\sum_{\{J_1,\ldots,J_{2(n-4)}\}\in\mathbf{NC}_{3,n}}\prod_{j=1}^{2(n-4)}\frac{1}{\hat{\eta}_{J_j}}.$$

This proposes a noncrossing-complex triangulation formula for the resolved amplitude, generalizing the corresponding associahedral amplitude formula; its resolution status is not established in the supplied text.