Ball's conjecture on the $SL(n)$-invariant quantity $B(K)$

For a convex body $K\subseteq\mathbb{R}^n$, define

$$B(K):=\int_K\int_{K^\circ}\langle x,y\rangle^2\,dx\,dy,$$

where $K^\circ$ is the polar body. Ball proved that $B(K)\leq B(B_2^n)$ for unconditional convex bodies. Ball's conjecture. The same inequality should hold for every symmetric convex body $K$ in $\mathbb{R}^n$:

$$B(K)\leq B(B_2^n).$$

The conjecture extends Ball's unconditional-body theorem to arbitrary symmetric convex bodies and is presented in the source as unresolved.