Alternation conjecture for Euler characteristics of triple Schubert-cell intersections

Let $G/P$ be a partial flag manifold, let $u,v,w\in W^P$, and let $Y(uW_P)^\circ$ and $X(wW_P)^\circ$ denote the corresponding opposite and ordinary Schubert cells. For general-position elements $g_1,g_2,g_3\in G$, consider the Euler characteristic of their intersection.

Alternation conjecture. The Euler characteristic is alternating:

$$(-1)^{\ell(u)+\ell(v)+\ell(w)} \chi\bigl(g_1 Y(uW_P)^\circ \cap g_2 Y(vW_P)^\circ \cap g_3 X(wW_P)^\circ\bigr) \ge 0.$$

The transversality theorem identifies these Euler characteristics with structure constants of products of Segre–MacPherson classes, making the conjecture a positivity statement for classical intersection-theoretic objects. Its general status is not resolved in the supplied text.