Fradelizi--Meyer's affine-invariant Mahler conjecture

Let $K\subset\mathbb{R}^n$ be a convex body. For $z\in\mathbb{R}^n$, define its polar with respect to $z$ by

$$K^z=\{y\in\mathbb{R}^n:\langle y-z,x-z\rangle\leq 1\text{ for all }x\in K\},$$

and define the affine-invariant volume product by

$$P(K)=\min_{z\in\mathbb{R}^n}|K||K^z|.$$

Let $\Delta^n$ be an arbitrary non-degenerate simplex in $\mathbb{R}^n$, not necessarily centered.

Fradelizi--Meyer's conjecture. For every convex body $K\subset\mathbb{R}^n$,

$$P(K)\geq P(\Delta^n)=\frac{(n+1)^{n+1}}{(n!)^2}.$$

Moreover, equality holds if and only if $K=\Delta^n$.

This is the affine-invariant formulation of the general Mahler conjecture and is equivalent to the centered formulation stated above. It remains open.