The first-n-coefficients conjecture for rational smoothness in simply laced root systems

Let $\Phi$ be a simply laced root system of rank $n$, with positive roots $\Phi_+$. For an element $w$ of its Weyl group, let $P_w(t)$ denote the corresponding Poincaré polynomial, and let $P_{\mathrm{id},w}(q)$ be the Kazhdan–Lusztig polynomial. The pair $(\Phi_+,w)$ is called not rationally smooth when the relevant rational smoothness criterion fails.

First-$n$-coefficients conjecture. If $(\Phi_+,w)$ is not a rationally smooth pair, then comparing the first $n$ coefficients and the last $n$ coefficients of $P_w(t)$ suffices to find an asymmetry. Equivalently, $P_{\mathrm{id},w}(q)$ has a nonzero coefficient among

$$q^1,q^2,\dots,q^n.$$

This gives a finite low-degree test for detecting rational singularity of Schubert varieties in simply laced types. The supplied text does not indicate whether the conjecture has been resolved.