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We determine all connected homogeneous Kobayashi-hyperbolic manifolds ofdimension $n\ge 4$ whose group of holomorphic automorphisms has dimensioneither $n^2-4$, or $n^2-5$, or $n^2-6$. This paper continues a series ofarticles that achieve classifications for automorphism group dimension $n^2-3$and greater.

Further steps towards classifying homogeneous Kobayashi-hyperbolic manifolds with high-dimensional automorphism group