Chamberland–Meisters injectivity conjecture for uniformly nonvanishing Jacobian eigenvalues

Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map. Chamberland–Meisters' injectivity conjecture. Suppose there exists an $\varepsilon>0$ such that

$$|\lambda|\geqslant\varepsilon$$

for every eigenvalue $\lambda$ of $F'(x)$ and every $x\in\mathbb{R}^n$. Then $F$ is injective. The conjecture is presented as a sufficient condition for the Jacobian conjecture; the supplied text does not establish whether it has been resolved.