On quasi-similarity of model contractions with non-equal defects

Let $T_\theta$ and $T_\Phi$ be $C_{10}$ contractions with characteristic functions $\theta\in H^\infty(\mathbb C^n\to\mathbb C^{n+1})$ and $\Phi\in H^\infty(\mathbb C^m\to\mathbb C^{m+1})$. The main result: $T_\theta$ and $T_\Phi$ are quasi-similar iff
$$ \{\det(f,\theta)^i:f\in H_n^2\}=\{\det(g,\Phi)^i:g\in H_m^2\}. $$
The paper contains an analysis of this condition. Some examples are given.