Mahler's successive-minima inequality for polars

Let $K\in\mathcal K_{os}^n$ be an origin-symmetric convex body and let $\Lambda\in\mathcal L^n$ be a lattice. Write $K^\star$ and $\Lambda^\star$ for the polar body and dual lattice. Mahler's successive-minima conjecture.

$$\frac{2^n}{n!}\det(\Lambda)\,\lambda_1(K^\star,\Lambda^\star)\cdots\lambda_n(K^\star,\Lambda^\star)\leq\operatorname{vol}(K).$$

The inequality would be best possible and, according to the source, is known for $n=2,3$; the source does not resolve the general case.