The volume-product conjecture for finite metric spaces

Let $M$ be a metric space with $n+1$ points. Let $\mathcal F(M)$ be its Lipschitz-free space, let $B_{\mathcal F(M)}$ be the unit ball of this $n$-dimensional space, and define

$$\mathcal P(M):=\mathcal P(B_{\mathcal F(M)}).$$

Volume-product conjecture for finite metric spaces. One has

$$\mathcal P(M)\geq \mathcal P(B_1^n).$$

Here $B_1^n$ is the cross-polytope, whose volume product equals that of the cube and is the conjectured minimum among symmetric convex bodies. This is presented as the paper's main conjecture for the volume products of Lipschitz-free spaces; no resolution is given in the supplied text.