Makai's lattice-width volume conjecture

Let $K\in\mathcal K^n$ be a full-dimensional convex body and let $\Lambda\in\mathcal L^n$. Write $K_s$ for the symmetrization of $K$, and $K^\star$ and $\Lambda^\star$ for the polar body and dual lattice. Makai's conjecture.

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

The source says that this weaker inequality remains open for $n\geq4$.