Positivity conjecture for the signed Euler characteristic of three Schubert cells

Let $G/P$ be the homogeneous variety with Schubert cells $X_\lambda^\circ$, indexed by $W^P$, and let $a^\nu_{\lambda,\mu}\in\mathbb{Z}$ be the SSM structure constants defined by

$$s_{SM}(X_\lambda^\circ)\cdot s_{SM}(X_\mu^\circ)=\sum_\nu a^\nu_{\lambda,\mu}\cdot s_{SM}(X_\nu^\circ).$$

For generic $g,h\in G$, choose $\nu'$ satisfying $\nu'\cdot W_P=w_0\nu\cdot W_P$, and set

$$d:=\dim\bigl(X_\lambda^\circ\cap gX_\mu^\circ\cap hX_{\nu'}^\circ\bigr).$$

Positivity conjecture for the signed Euler characteristic. One has

$$E_{\lambda,\mu,\nu'}:=(-1)^d\cdot a^\nu_{\lambda,\mu}=(-1)^d\cdot\chi\bigl(X_\lambda^\circ\cap gX_\mu^\circ\cap hX_{\nu'}^\circ\bigr)\geq 0.$$

This conjecture predicts the positivity of the signed Euler characteristics arising from intersections of three generically translated Schubert cells, equivalently the corresponding SSM structure constants. It was formulated in work and talks of Mihalcea and Knutson and remains unresolved in the supplied context.