Volumetric zone conjecture on spherical zone unions

Let $d\geq 2$, let $u_1,\dots,u_n\in\mathbb{S}^{d-1}$, and let $\alpha\in[0,\pi/2]$. For a unit vector $u$, define the spherical zone

$$Z(u,\alpha)=\{v\in\mathbb{S}^{d-1}:|\langle v,u\rangle|\leq\sin(\alpha)\}.$$

For $j=1,\dots,n$, set

$$v_j=\left(\cos\left(\frac{(j-1)\pi}{n}\right),\sin\left(\frac{(j-1)\pi}{n}\right),0,\dots,0\right)\in\mathbb{S}^{d-1}.$$

Let $\sigma_{d-1}$ be normalized surface measure on $\mathbb{S}^{d-1}$. Volumetric zone conjecture.

$$\sigma_{d-1}\left(\bigcup_{j=1}^n Z(u_j,\alpha)\right)\leq\sigma_{d-1}\left(\bigcup_{j=1}^n Z(v_j,\alpha)\right).$$

The claim says that evenly spaced zones whose defining great subspheres share a codimension-two subsphere maximize the union's surface measure. The source relates this to Fejes Tóth's zone conjecture, whose covering version was proved by Jiang and Polyanskii; the volumetric extension stated here is presented as the unproved geometric reduction of the cosine covariance conjecture.