Integral Fejes Tóth conjecture for projective spaces

Let $\mathbb S^d$ be the unit sphere, let $\theta(x,y)$ be the geodesic angle, and for a Borel probability measure $\mu$ define

$$I_1(\mu)=\int_{\mathbb S^d}\int_{\mathbb S^d}\theta(x,y)\,d\mu(x)\,d\mu(y).$$

For an orthonormal basis $\{e_i\}_{i=1}^{d+1}$ of $\mathbb R^{d+1}$, set

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

Integral Fejes Tóth conjecture. The maximum of $I_1(\mu)$ over all Borel probability measures on $\mathbb S^d$ is achieved by $\mu_{ONB}$. The discrete conjecture remains open for $d>1$, and this continuous analogue is likewise not established in the stated generality; in projective-space terminology it is equivalent to the phase-transition assertion $s^*=-1$ for $\mathbb{RP}^d$.