Conjecture on uniform spectral separation implying injectivity

Let $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a $C^1$ map. Suppose there exists an $\varepsilon>0$ such that $|\lambda|\geqslant\varepsilon$ for every eigenvalue $\lambda$ of $F'(x)$ and every $x\in\mathbb{R}^n$. Uniform spectral injectivity conjecture. Under these assumptions, $F$ is injective. The result is stated as a higher-dimensional extension of the two-dimensional injectivity theorem discussed in the paper; the surrounding text says that the claim is true when $n=2$, while its status in general dimension is not specified here.