The "most symmetric" Mukai-Umemura 3-fold with automorphism group $PGL(2,\mathbb{C})$ admits a Kaehler-Einstein metric according to Donaldson's result.

On the contrary, there are some arbitrarily small complex deformations of the above $3$-fold which do not admit Kaehler-Einstein metrics, as shown by Tian. All examples considered by Tian seem to have no symmetries at all. Is it possible to find similarly arbitrarily small complex deformations with $\mathbb{C}^*$-action and which do not admit any Kaehler-Einstein metric ?