Quaternion-Sasakian manifolds and special holonomy Sasakian manifolds

Two well-known slogans are

A Sasakian manifold is the odd dimensional analogue of a Kähler manifold

and

A $$3$$-Sasakian manifold is the odd dimensional analogue of a hyper-Kähler manifold

Does this analogy extend to quaternion-Kähler manifolds? Is there a notion of quaternion-Sasakian manifolds?

What about special holonomy? Does there exist a notion of $$Spin(7)$$-Sasakian manifold. What about $$G_2$$-holonomy?

Can one use find an even dimensional analogue?

Nearly-Kahler manifold are the 6-dimensional links of G2-cones, see https://en.wikipedia.org/wiki/Nearly_K%C3%A4hler_manifold

I do not know about the others

• There is a similar result between dimensions $7$ and $8$ too. Apr 23, 2021 at 12:14