We know from the classy work of Joyce that "any compact Lie group becomes hypercomplex after it is multiplied by a sufficiently big torus". The quote comes from the Wikipedia [page](https://en.wikipedia.org/wiki/Hypercomplex_manifold).


I am asking if it is known that these hypercomplex manifolds are hyper-Kaehler (or hyper-Hermitian) or not.



Hyper-Kaehler Strucutre for Compact Lie Groups?
The answer is already 'no' for the simplest case:  

$$
S^1\times \mathrm{SU}(2) = (\mathbb{H}{\setminus}\{0\})/\mathbb{Z},
$$
which is clearly hypercomplex, but cannot even be Kähler, much less hyperKähler.


