On the minimal power of q in a Kazhdan–Lusztig polynomial

For w in the symmetric group, we provide an exact formula for the smallest positive power $q^{h\left(w\right)}$ appearing in the Kazhdan–Lusztig polynomial $P_{e,w}\left(q\right)$. We also provide a tight upper bound on $h\left(w\right)$ in simply-laced types, resolving a conjecture of Billey–Postnikov from 2002.