The finiteness conjecture for homogeneous Einstein metrics

Let $M=\mathsf G/\mathsf H$ be a compact homogeneous space whose isotropy representation consists of pairwise inequivalent irreducible summands. In particular, this includes the case $\operatorname{rank}\mathsf G=\operatorname{rank}\mathsf H$. Finiteness conjecture. The Einstein equations have only finitely many real solutions. This conjecture addresses the general finiteness problem for homogeneous Einstein metrics; the results preceding it give sufficient, but not necessary, algebraic conditions for finiteness in the setting of pairwise inequivalent isotropy summands.