Unit-cube conjecture for the symmetric convex extremal

Let $\mathcal{SC}$ be the class of bounded convex domains symmetric with respect to each coordinate axis, and let

$$M_{p,d}(\mathcal{SC})=\sup_{D\in\mathcal{SC}}G_{p,d}(D).$$

For $p>0$, define the unit cube

$$Q_d=\{(x_1,\ldots,x_d)\in\mathbb{R}^d:|x_i|<1\}.$$

Unit-cube conjecture.

$$M_{p,d}(\mathcal{SC})=\lambda_1(Q_d)^p\mathbb{E}_0[\tau_{Q_d}^p].$$

The paper proves existence of an extremal in this class, but the assertion that the unit cube is an extremal is stated as a conjecture and is not proved in the supplied text.