Quotient spaces modulo tori and transitive actions of Lie groups

It is proved that a transitive locally effective action of a compact connected simply-connected Lie group on a manifold of the form $K/A$, where $K$ is a non-simple compact connected simply-connected Lie group all of whose simple factors have rank greater than 5 and $A$ is a torus in $K$ in general position of corank 1, is equivalent to the original action of the group $K$.