Woo–Billey–Weed conjecture on low-degree Kazhdan–Lusztig polynomials

Let $w\in S_n$, let $P_{\mathrm{id},w}(q)$ be the Kazhdan–Lusztig polynomial, and let $\operatorname{ms}(w)$ denote the set of permutations indexing the irreducible components of the singular locus of the Schubert variety $X_w$. Woo–Billey–Weed's conjecture.

1. If $P_{\mathrm{id},w}(1)\leq 3$, then $|\operatorname{ms}(w)|\leq 3$.
2. If $P_{\mathrm{id},w}(1)=3$ and $|\operatorname{ms}(w)|=1$, then $P_{\mathrm{id},w}(q)=1+q^a+q^b$ for some $0<a<b$.
3. If $P_{\mathrm{id},w}(1)=3$ and $|\operatorname{ms}(w)|=2$, then $P_{\mathrm{id},w}(q)=1+q^a+q^b$ for some $0<a<b$.
4. If $P_{\mathrm{id},w}(1)=3$ and $|\operatorname{ms}(w)|=3$, then $P_{\mathrm{id},w}(q)=1+2q^a$ for some $0<a$.

The conjecture had been tested through $S_8$ according to the supplied context, and the source presents it among open problems in Kazhdan–Lusztig theory.