  We completely characterize the pairs of connected Lie groups $G > K$ such
that $\mathrm{rank}(G) - \mathrm{rank}(K) = 1$ and the left action of $K$ on
$G/K$ is equivariantly formal. The analysis requires us to correct and extend
an existing partial classification of homogeneous quotients $G/K$ with the
rational homotopy type of a product of an odd- and an even-dimensional sphere.
