Caselli's nonnegativity conjecture for Kazhdan–Lusztig coefficients

Let $[u,v]$ be a Bruhat interval, let $n=l(u,v)-1$, and let $a_i(u,v)$ denote the coefficients in the expansion of its Kazhdan–Lusztig polynomial in the basis $q^i B_{n-2i}(-q)$.

Caselli's conjecture. For each Bruhat interval $[u,v]$ and each $i=0,1,\dots,\lfloor n/2\rfloor$, one has

$$a_i(u,v)\ge 0,$$

where $n=l(u,v)-1$.

These inequalities refine the conjectured coefficientwise nonnegativity of Kazhdan–Lusztig polynomials and can be interpreted, via the complete $\mathbf{cd}$-index, as a system of linear inequalities for its coefficients. The source provides no resolution of the conjecture.