  We prove that for every $n\geq 3$ the sharp upper bound for the dimension of
the symmetry groups of homogeneous, 2-nondegenerate, $(2n+1)$-dimensional CR
manifolds of hypersurface type with a $1$-dimensional Levi kernel is equal to
$n^2+7$, and simultaneously establish the same result for a more general class
of structures characterized by weakening the homogeneity condition. This
supports Beloshapka's conjecture stating that hypersurface models with a
maximal finite dimensional group of symmetries for a given dimension of the
underlying manifold are Levi nondegenerate.
