The volume lower bound for real reflexive polytopes

Let $P$ be a real reflexive polytope, meaning a reflexive polytope for which every root of its Ehrhart polynomial $L_P$ is real. Let $\operatorname{vol}(P)$ denote its normalized volume.

Real reflexive-polytope volume conjecture. If $P$ is $d$-dimensional, then

$$\operatorname{vol}(P)\geq 2^d.$$

The paper proves this claim in dimension $4$ using the classification of Kreuzer and Skarke, but notes that no theoretical explanation is known there; the general-dimensional conjecture remains open in the supplied text.