Hypersimplex Ehrhart-root strip conjecture

Let $d$ and $n$ be integers with $1\leq d<n$, and let $\Delta(d,n)$ be the convex hull in $\mathbb{R}^n$ of the vectors $\mathbf e_{i_1}+\cdots+\mathbf e_{i_d}$ with $1\leq i_1<\cdots<i_d\leq n$. Assume $2d\leq n$. The Ehrhart polynomial $i(\Delta(d,n),m)$ counts the integer points in $m\Delta(d,n)$. Hypersimplex Ehrhart-root strip conjecture. Every root $a\in\mathbb{C}$ of $i(\Delta(d,n),m)$ satisfies

$$-\frac{n}{d}<\operatorname{Re}(a)<0.$$

The claim is supported by the known case $d=2$ and computational experiments. The paper proves it for $d=3$ and establishes related bounds when $4\leq d\ll n$; the conjecture is therefore open in the remaining cases described by the source.