Are there examples of Fano manifolds such that Tian's alpha invariant satisfies $\alpha_G(X)=\frac{n}{n+1}$ but without a Kähler-Einstein metric?

The alpha invariant $\alpha(X)$ of a Fano manifold $X$ of dimension $n$ is defined as the infimum of log canonical thresholds of (effective) $\mathbb{Q}$-divisors $D\sim_{\mathbb{Q}}-K_X$. Similarly, for $G\subset Aut(X)$ a compact subgroup of the automorphism group, one defines $\alpha_G(X)$ considering only $G$-invariant divisors. The alpha invariant has a corresponding analytic definition involving complex singularity exponents of singular hermitian metrics [2, Appendix].

Tian introduced this invariant and proved that the lower bound $\alpha_G(X)>\frac{n}{n+1}$ implies the existence of a Kähler-Einstein metric  (in fact, even today it is one of the few sufficient conditions which is checkable in practice). I'd like to know if this theorem is sharp? That is:

Question: Are there examples Fano manifolds such that $\alpha_G(X)=\frac{n}{n+1}$ but without a Kähler-Einstein metric?

I'd also be interested in any partial results in the positive direction.

An example I know of with $\alpha(X)=\frac{n}{n+1}$ is a del Pezzo surface of degree $4$ (this is due to Cheltsov ), however by Tian's classification of Kähler-Einstein metrics on del Pezzo surfaces , such surfaces are known to admit Kähler-Einstein metrics.

References:

 G. Tian. On Kahler-Einstein metrics on certain K ̈ahler manifolds with $c_1(M)>0$.

 I. Cheltsov, C. Shramov, Appendix by J. P. Demailly. Log canonical thresholds of smooth Fano threefolds.

 I. Cheltsov. Log canonical thresholds of del Pezzo surfaces.

 Tian, G. On Calabi’s conjecture for complex surfaces with positive first Chern class.

• Can you please give a reference for this statement? Thanks a lot. – ACL Aug 14 '13 at 17:41
• No problem, I'll add some references. – Ruadhaí Dervan Aug 14 '13 at 18:00

This question was answered negatively by Kento Fujita today (at least when $G$ is trivial).
Theorem (Fujita): If $\alpha(X,-K_X)=\frac{n}{n+1}$, then $X$ is K-stable and hence admits a Kähler-Einstein metric.
• I don't think this answers the question, which was for examples where $\alpha = \frac n {n+1}$ and no Kähler-Einstein metric exists. – Dan Petersen Aug 15 '13 at 7:15
• Sorry Vanya, perhaps I should have been more clear. Dan is right, what I'd like is an example where $\alpha_G(X)=\frac{n}{n+1}$ but where no Kähler-Einstein metric exists. I'll edit the question to make it clearer. Thanks for the answer, though. – Ruadhaí Dervan Aug 15 '13 at 10:26