  We show that the tilted perfect fluid Bianchi VI$_0$ family of self-similar
models found by Rosquist and Jantzen [K. Rosquist and R. T. Jantzen, \emph{%
Exact power law solutions of the Einstein equations}, 1985 Phys. Lett.
\textbf{107}A 29-32] is the most general class of tilted self-similar models
but the state parameter $\gamma $ lies in the interval $(\frac 65,\frac 32) $.
The model has a four dimensional stable manifold indicating the possibility
that it may be future attractor, at least for the subclass of tilted Bianchi
VI$_0$ models satisfying $n_\alpha ^\alpha =0$ in which it belongs. In addition
the angle of tilt is asymptotically significant at late times suggesting that
for the above subclasses of models the tilt is asymptotically extreme.
