Kazhdan–Lusztig coefficient maximality conjecture for uniform matroids

Let $M$ be a matroid of rank $d$ on $m+d$ elements, and write its Kazhdan–Lusztig polynomial as

$$P_M(t)=\sum_i c^i t^i.$$

Let $U_{m,d}$ denote the uniform matroid of rank $d$ on $m+d$ elements, and let $c^i_{m,d}(\emptyset)$ be the $i$-th coefficient of its Kazhdan–Lusztig polynomial. Kazhdan–Lusztig coefficient maximality conjecture. For every such matroid $M$ and every coefficient index $i$,

$$c^i\leq c^i_{m,d}(\emptyset).$$

Equivalently, among matroids with rank $d$ and ground-set size $m+d$, the uniform matroid has the largest Kazhdan–Lusztig coefficients. The paper proves this conjecture for sparse paving matroids, but the general statement remains open.