Ball's Santaló type inequality for symmetric convex bodies

Let $K$subseteq$\mathbb R^n$ be a symmetric convex body, and define its polar body by

$$K^\circ=\{x\in\mathbb R^n:\langle x,y\rangle\leq 1\;\forall y\in K\}.$$

Let $B_2^n=\{x\in\mathbb R^n:\|x\|\leq 1\}$ be the Euclidean unit ball. Ball's conjecture.

$$\int_K\int_{K^\circ}\langle x,y\rangle^2\,dx\,dy\leq\int_{B_2^n}\int_{B_2^n}\langle x,y\rangle^2\,dx\,dy.$$

This is a Santaló-type inequality comparing a symmetric convex body with its polar, with the Euclidean ball as the extremal case. The conjecture is resolved by the paper's proof; the abstract also states a stronger isotropic directional inequality and characterizes equality.