Billey–Fan–Losonczy's Lehmer-code conjecture for rationally smooth elements

Let $W$ be a finite Weyl group, let $w\in W$ be rationally smooth, and let $[e,w]$ be its Bruhat interval. A Lehmer code is an order-preserving bijection from a product of chains to $[e,w]$.

Billey–Fan–Losonczy's conjecture. The interval $[e,w]$ admits a Lehmer code.

Gasharov proved the corresponding statement for $W=S_n$. The conjecture has been resolved affirmatively in types $A$ and $B$, and the paper resolves it in all types except $E$; in types $H_4$ and $F_4$, the longest element is the stated exception.