Billey and Postnikov's bound on the minimal Kazhdan–Lusztig degree

Let $G$ be a simply-laced algebraic group of rank $r$, with Weyl group $W$, and let $X_w\subset G/B$ be the Schubert variety associated to $w\in W$. For $w\in W$, define

$$h(w)=\min\{i>0\mid [q^i]P_{e,w}(q)\neq 0\},$$

where $P_{e,w}(q)$ is the Kazhdan–Lusztig polynomial; set $h(w)=+\infty$ when $X_w$ is smooth.

Billey and Postnikov's conjecture. If $X_w\subset G/B$ is singular, then

$$h(w)\leq r.$$

The quantity $h(w)$ is the first degree in which the Kazhdan–Lusztig polynomial has a nonzero coefficient, equivalently the first degree in which local intersection cohomology detects failure of local Poincaré duality. The conjecture gives a rank-dependent upper bound for this first nontrivial degree in singular Schubert varieties.