Finiteness conjecture for cohomogeneity-one Einstein metrics on even-dimensional spheres

An even-dimensional sphere is a sphere $S^n$ with $n$ even; a non-round Einstein metric is an Einstein metric that is not isometric to the standard round metric.

Finiteness conjecture. Only finitely many even-dimensional spheres admit non-round Einstein metrics of cohomogeneity one.

The conjecture is motivated by the contrast between the three known metrics on $S^{10}$ and the existence of infinitely many such metrics on $S^6$ and $S^8$. It remains open in the supplied source.