Multi-bubble isoperimetric conjecture on the sphere

Let $q$ satisfy $2\leq q\leq n+2$, and let $\Omega$ be a $q$-cluster on $\mathbb{S}^n$ with prescribed volume $V(\Omega)=v\in\operatorname{int}\Delta^{(q-1)}_{1}$. A standard bubble is the canonical spherical cluster whose interfaces lie on great or generalized spheres and meet three at $120^\circ$ angles. Multi-bubble isoperimetric conjecture. For all $2\leq q\leq n+2$, a standard bubble uniquely minimizes total perimeter among all such $q$-clusters. This is the spherical multi-bubble isoperimetric problem; the claim remains open in the stated generality.