The discriminant-sign conjecture for monotone Grassmannian Schubert problems

Let $(w_1,\dotsc,w_m)$ be a Grassmannian Schubert problem for $\mathbb{F}\ell(\alpha;n)$, let $\delta(w_i)$ be the unique descent of $w_i$, and let $\Delta_w(t_1,\dotsc,t_m)$ be its discriminant polynomial. Define

$$S=\{t_i-t_j\mid \delta(w_i)>\delta(w_j)\}.$$

The preorder generated by $S$ consists of sums $\sum_\varepsilon c_\varepsilon\prod_i g_i^{\varepsilon_i}$, where $g_i\in S$, $\varepsilon_i\in\{0,1\}$, and each $c_\varepsilon$ is a sum of squares of polynomials. Discriminant-sign conjecture. The discriminant $\Delta_w$, or its negative, lies in the preorder generated by $S$. Such membership would make the sign of the discriminant on monotone points evident and would imply nonvanishing there; the conjecture is presented as open and is supported by the computed examples.