The $C^1$ stability conjecture

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map, and let $J(f)$ denote its Jacobian matrix. The $C^1$ stability conjecture. If the eigenvalues of $J(f)$ have strictly negative real part at every point, then $f$ is injective. The paper discusses this in connection with the Markus–Yamabe conjecture and notes that the polynomial stability conjecture would imply the Jacobian Conjecture; the $C^1$ statement remains unresolved in the source.