Chamberland–Meisters eigenvalue conjecture for injectivity

Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map. Suppose there exists an $\varepsilon>0$ such that, for every $x\in\mathbb{R}^n$ and every eigenvalue $\lambda$ of $F'(x)$,

$$|\lambda|\geq\varepsilon.$$

Chamberland–Meisters conjecture. Then $F$ is injective. This conjecture is presented as a sufficient condition for injectivity after the real Jacobian conjecture was disproved; its resolution is not given in the supplied text.