  We extend the results of arXiv:2206.08295v2 by showing that any homothety in
$\mathbb T^2$ is homotopic to a non-uniformly hyperbolic ergodic area
preserving map, provided that its degree is at least $5^2$. We also address
other small topological degree cases not considered in the previous article.
This proves the existence of a $\mathcal C^1$ open set of non-uniformly
hyperbolic systems, that intersects essentially every homotopy classes in
$\mathbb T^2$, where the Lyapunov exponents vary continuously.
