Makai Jr.'s non-symmetric difference-body volume-product conjecture

Let $K$ be a convex body in $\mathbb{R}^{n}$, and let $(K-K)^{\circ}$ be the polar body of its difference body. Makai Jr.'s conjecture.

$$|K|\,|(K-K)^{\circ}| \geq \frac{n+1}{n!}.$$

Equality should hold if and only if $K$ is a simplex. This alternative non-symmetric extension is intended to preserve the connection between Mahler-type inequalities, geometry of numbers, and non-separable lattice arrangements; the source does not state a resolution status.