Makai Jr.'s covering-minimum conjecture

Let $K\in\mathcal{K}^n$ be a convex body and let $\Lambda\in\mathcal{L}^n$ be a lattice. Let $\mu_1(K,\Lambda)$ denote the first covering minimum. Makai Jr.'s conjecture. One has

$$\mu_1(K,\Lambda)^n\operatorname{vol}(K)\geq\frac{n+1}{2^n n!}\det(\Lambda),$$

and equality can occur only when $K$ is a simplex. If $K$ is origin-symmetric, then

$$\mu_1(K,\Lambda)^n\operatorname{vol}(K)\geq\frac{1}{n!}\det(\Lambda),$$

and equality can occur only when $K$ is a crosspolytope. This conjecture is presented as a polar analogue of Minkowski's first fundamental theorem and remains unresolved in the source.