Dimension conjecture: in search of symmetry

In two-dimensional complex space there is the alternative: either the Lie algebra of infinitesimal holomorphic automorphisms of the germ of a CR manifold has infinite dimension, or its dimension does not exceed eight, and the maximum is attained at the three-dimensional sphere $v=|z|^2$. For a long time the following question in CR geometry was open: is it true that the most symmetrical objects are the generalizations of the sphere to higher CR dimensions and codimensions — nondegenerate model surfaces? Recently, it was discovered that the answer is negative: explicit counterexamples were found. For any CR type $(n,k)$ with the condition $k>1$ and for any (arbitrarily large) number $m$ there exist a germ of a manifold of type $(n,k)$, which have a finite dimensional automorphisms algebra of dimension greater than $m$. We will discuss, how to correct the formulation of the dimension conjecture, and state some unsolved problems.