Bishop's conjecture for the minimal 3-partition of the sphere

Let $\Gamma$ be the sphere and consider the problem of minimizing the sum of the first Dirichlet eigenvalues over partitions of $\Gamma$ into three parts. A Y-partition is the partition whose boundary, up to a fixed rotation, is the intersection of $\Gamma$ with the three half-planes defined in polar coordinates by $\phi=0$, $\phi=\frac{2\pi}{3}$, and $\phi=-\frac{2\pi}{3}$. Bishop's conjecture. The minimal 3-partition is this Y-partition. The source presents this as the spherical $m=3$ case and refers to it as the Bishop conjecture; the surrounding discussion gives the claimed geometric configuration but does not state a resolution.