The antipodal isoperimetric conjecture for the sphere

Let $n \in \mathbb{N}$ with $n \geq 2$, and let $\mu$ denote the $n$-dimensional Hausdorff measure on $\mathbb{S}^n$. Let $\mathcal{A} \subseteq \mathbb{S}^n$ be open and antipodal if $\mathcal{A}=-\mathcal{A}$. Let $\sigma$ denote the boundary-size functional used in the isoperimetric problem. The antipodal isoperimetric conjecture. There exists a set $\mathcal{B} \subseteq \mathbb{S}^n$ such that $\mu(\mathcal{B})=\mu(\mathcal{A})$, $\sigma(\mathcal{B})\leq\sigma(\mathcal{A})$, and

$$\mathcal{B}=\left\{x\in\mathbb{S}^n:\sum_{i=1}^{r}x_i^2>a\right\}$$

for some $r\in[n]$ and some $a\in\mathbb{R}$. This is the isoperimetric problem for real projective space, equivalently for antipodal subsets of the sphere; it remains open in general.