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In this paper we prove that the homotopy class of non-homothety linearendomorphisms on $\mathbb{T}^2$ with determinant greater than 2 contains a$C^1$ open set of non-uniformly hyperbolic endomorphisms. Furthermore, we provethat the homotopy class of non-hyperbolic elements (having either $1$ or $-1$as an eigenvalue) whose degree is large enough contains non-uniformlyhyperbolic endomorphisms that are also $C^2$ stably ergodic. These resultsprovide partial answers to certain questions posed in arXiv:2206.08295v2

Non-uniform hyperbolicity of maps on $\mathbb{T}^2$