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Any $6$-dimensional strict nearly K\"ahler manifold is Einstein with positivescalar curvature. We compute the coindex of the metric with respect to theEinstein-Hilbert functional on each of the compact homogeneous examples.Moreover, we show that the infinitesimal Einstein deformations on$F_{1,2}=\mathrm{SU}(3)/T^2$ are not integrable into a curve of Einsteinmetrics.

Coindex and rigidity of Einstein metrics on homogeneous Gray manifolds