Weight-Two Classification for positive tropical Grassmannians

Let $I,J\in {\binom{[n]}{k}}^{\mathrm{ncyc}}$ be noncrossing but not weakly separated. For each such $I$, let $\mathfrak{t}_I$ denote the corresponding noncrossing parameter, and let $\rho$ be the positive parameterization into the positive tropical Grassmannian.

Weight-Two Classification. The point

$$\rho(\mathfrak{t}_I+\mathfrak{t}_J)$$

lies in the direction of a ray of $\operatorname{Trop}_{>0}X(k,n)$. Every ray of $\operatorname{Trop}_{>0}X(k,n)$ with PK weight $2$ arises in this way from a unique noncrossing but not weakly separated pair $\\{I,J\\}$.

This conjecture gives the proposed classification of the weight-two rays of the positive tropical Grassmannian, extending the preceding classification of weight-one rays. It was stated previously in the cited work, but its resolution is not specified in the supplied context.