The $d$-sequence conjecture for coefficients of uniform-matroid $Z$-polynomials

For integers $m$ and $d$, and $0\leq i\leq d$, define

$$b_m(d,i)=\sum_{h=0}^{m-1}\frac{i(h-m+1)+m}{(h+1)m}\binom{i-1+h}{h}\binom{d-i+h}{h}.$$

A finite sequence of real numbers is an $n$-sequence when the associated polynomial has real zeros of one common sign. The $d$-sequence conjecture. The sequence $\{b_m(d,i)\}_{i=0}^{d}$ is a $d$-sequence for every $m$ and $d$. The conjecture is proposed as the analogue of the corresponding assertion for Kazhdan–Lusztig polynomials and would support real-rootedness results for uniform-matroid $Z$-polynomials; its general status remains open.