Kuperberg's bottleneck conjecture

Let $V$ be an $n$-dimensional vector space, let $V^*$ be its dual, and let $K\subset V$ be a centrally symmetric convex body centered at the origin. Write

$$K^+=\{(\vec{x},\vec{y})\in K\times K^\circ\mid\langle\vec{x},\vec{y}\rangle=1\},$$ $$K^-=\{(\vec{x},\vec{y})\in K\times K^\circ\mid\langle\vec{x},\vec{y}\rangle=-1\},$$

and let $K^\diamondsuit$ be the convex hull of $K^+\cup K^-$. Define

$$D(K)=\operatorname{Vol}K^\diamondsuit.$$

Bottleneck conjecture. For convex bodies $K$ in $n$ dimensions with $n$ fixed, the volume $D(K)$ is uniquely minimized when $K$ is an ellipsoid.

The inclusion $K^\diamondsuit\subseteq K\times K^\circ$ gives $D(K)\leq M(K)$, and the conjecture would improve the best asymptotic lower bound on Mahler volume described in the source. The source does not report a resolution.