Singleton conjecture for proper gradient homotopy classes of planar maps of negative degree

Let $\mathcal{P}_{k}^\nabla[\mathbb{R}^{2}]$ denote the set of proper gradient homotopy classes of proper maps $\mathbb{R}^{2}\to\mathbb{R}^{2}$ of degree $k$. Singleton conjecture. For every integer $k<0$, $\mathcal{P}_{k}^\nabla[\mathbb{R}^{2}]$ is a singleton. This complements the paper's results for nonnegative degrees and predicts uniqueness of the proper gradient homotopy class in every negative degree; the statement is presented as a conjecture and remains open here.