Beck–Stapledon conjecture on the real-rootedness of dilated Ehrhart $h^*$-polynomials

Let $P$ be a $d$-dimensional lattice polytope, and let $h^*(rP)$ denote the $h^*$-polynomial of its dilation $rP$. Beck–Stapledon conjecture. The polynomial $h^*(rP)$ has only distinct, negative, real zeros for all $r\geq d$. This conjecture gives an optimal bound for the onset of real-rootedness in the Ehrhart setting, strengthening the known fact that sufficiently large dilations have only real roots.