Fejes Tóth's continuous energy conjecture on the sphere

Let $\mathcal B(\mathbb S^d)$ be the set of Borel probability measures on $\mathbb S^d$, and for $\mu\in\mathcal B(\mathbb S^d)$ define

$$I(\mu)=\int_{\mathbb S^d}\int_{\mathbb S^d}\arccos\lvert x\cdot y\rvert\,d\mu(x)d\mu(y).$$

Let

$$\nu_{ONB}=\frac{1}{d+1}\sum_{i=1}^{d+1}e_i.$$

Fejes Tóth's continuous energy conjecture. The energy integral is maximized by $\nu_{ONB}$:

$$\max_{\mu\in\mathcal B(\mathbb S^d)}I(\mu)=I(\nu_{ONB})=\frac{\pi}{2}\cdot\frac{d}{d+1}.$$

This is the continuous analogue of the discrete conjecture. The supplied status evidence says that only the case $d=1$ of the discrete conjecture has been settled; no resolution of this continuous assertion is given, so it remains open.