Nonextendability of the dominated splitting at the singularities

Let $\Lambda$ be the singular attractor containing the hyperbolic singularities $s_0$ and $s_1$ of different indexes. On the part of $\Lambda$ away from the equilibria, suppose a dominated splitting is defined. Nonextendability conjecture. It is not possible to extend this dominated splitting on $\Lambda$ away from equilibria to the equilibria $s_0,s_1$ which belong to $\Lambda$. The different stable dimensions of the two singularities obstruct extending the dominated splitting across them.