On an extension of the Blaschke-Santalo inequality

Let $K$ be a convex body and $K^{\circ }$ its polar body. Call $\varphi (K)=\frac{1}{\mathrm{\mid }K\mathrm{\mid }\mathrm{\mid }{K}^{\circ }\mathrm{\mid }}{\int }_K{\int }_{K^{\circ }}<x,y{>}^{2}dxdy$. It is conjectured that $\varphi (K)$ is maximum when $K$ is the euclidean ball. In particular this statement implies the Blaschke-Santalo inequality. We verify this conjecture when $K$ is restricted to be a $p$–ball.