The nonreal-root real-part bound in dimensions six and seven

Let $P$ be a $6$- or $7$-dimensional reflexive polytope, and let $L_P$ be its Ehrhart polynomial. Suppose that $L_P$ has a root of the form

$$-\frac{1}{2}+a+bi\in\mathbb{C},$$

where $a\neq 0$ and $b\neq 0$.

Nonreal-root real-part conjecture. Then

$$|a|<\frac{5}{2}.$$

This is presented as an experimentally suggested conjecture after the paper's classifications in dimensions six and seven; no proof or resolution is given in the supplied text.