The volume lower-bound conjecture for 0-symmetric lattice polytopes

Let $\mathcal{P}_o^n$ denote the class of $0$-symmetric lattice polytopes in $\mathbb{R}^n$, let $\operatorname{int} P$ be the interior of $P$, let $\mathrm{G}(\operatorname{int} P)$ be the number of lattice points in its interior, and let $\operatorname{vol}(P)$ denote its volume.

Volume lower-bound conjecture. Every $P\in\mathcal{P}_o^n$ satisfies

$$\operatorname{vol}(P)\geq \frac{2^{n-1}}{n!}\left(\mathrm{G}(\operatorname{int} P)+1\right).$$

This is proposed as the $0$-symmetric analogue of the preceding sharp volume lower bound for general lattice polytopes. The surrounding text gives sharp upper bounds and examples for the symmetric case, but provides no resolution of this lower-bound assertion.