The covering-product lower-bound conjecture for convex bodies

For a convex body $K\in\mathcal{K}^n$, let $\mu_i(K)$ denote its $i$th covering minimum. Covering-product lower-bound conjecture. For every $K\in\mathcal{K}^n$,

$$\mu_1(K)\cdots\mu_n(K)\operatorname{vol}(K)\geq\frac{n+1}{2^n},$$

with equality, for example, for

$$T_n=\operatorname{conv}\{e_1,\ldots,e_n,-\mathbf{1}\}.$$

The conjecture seeks the sharp lower bound for the covering product; the paper notes that its value should decay exponentially with dimension and verifies the bound in dimension two and for important classes of bodies, while the general case remains open.