Álvarez Paiva et al.'s equality conjecture for lattice width

Let $K\in\mathcal K_o^n$ be an origin-centered convex body and let $\Lambda\in\mathcal L^n$. Write $K^\star$ and $\Lambda^\star$ for the polar body and dual lattice. Álvarez Paiva et al.'s conjecture.

$$\frac{n+1}{n!}\det(\Lambda)\,\lambda_1(K^\star,\Lambda^\star)^n\leq\operatorname{vol}(K),$$

with equality if and only if $K$ is a simplex whose only non-trivial lattice points are its vertices. This is presented as an interesting weaker inequality; the source does not give a resolution.