Several days ago I asked in
counterexample to the Chern number inequality on Fano manifold
that whether there exists an $n$-dimensional Fano manifold such that it does not satisfy Yau's Chern number inequality $nc_1^n\leq2(n+1)c_2c_1^{n-2}$.
Then abx provided the desired example. This example and its various generalizations can also be found in Olivier Debarre's book "Higher-dimensional algebraic geometry" (page 137).
The automorphism group of this example is not reductive (page 138 of this book) and so the property that it does not have any K-E metric can also be derived from Matsushima's theorem.
So my question is whether there exists an example whose automorphism group is reductive such that it does not satisfy Yau's Chern number inequality.
Thanks in advance.
