The ellipsoidal maximization conjecture for the polar second moment

Let $K\subseteq\mathbb{R}^n$ be centrally symmetric, let $K^\circ$ be its polar body, and let $(\vec{x},\vec{y})$ be uniformly distributed on $K\times K^\circ$. Polar second-moment conjecture. The quantity

$$E_{K\times K^\circ}[(\vec{x}\cdot\vec{y})^2]$$

is maximized when $K$ is an ellipsoid.

The conjecture would imply the centrally symmetric isotropic constant conjecture and would sharpen the Blaschke–Santaló theorem. The paper therefore describes it as difficult and perhaps less likely, but leaves it open.