Positivity conjecture for structure coefficients of the matroid Kazhdan–Lusztig basis

Let $M$ be a matroid with lattice of flats $L(M)$, let $E_q(M)$ be its $q$-deformed Möbius algebra, and let $\{x_F\mid F\in L(M)\}$ be the Kazhdan–Lusztig basis. Define structure coefficients $C_{FG}^H(q)\in\mathbb{Z}[q]$ by

$$x_F\cdot x_G=\sum_H C_{FG}^H(q)x_H.$$

Structure-coefficient positivity conjecture. For all $F,G,H\in L(M)$, $C_{FG}^H(q)\in\mathbb{N}[q]$. The conjecture was disproved by examples in rank $4$ and higher, including a uniform matroid of rank $4$ on $6$ elements.