Chamberland's injectivity conjecture for mappings with bounded inverse spectral radius

Let $n\geqslant 2$ and let $f:\mathbb{R}^n\to\mathbb{R}^n$ be continuously differentiable. Assume that $\det f'(x)\neq 0$ for every $x$ and that

$$\rho\bigl(f'(x)^{-1}\bigr)\leqslant A$$

for all $x\in\mathbb{R}^n$, where $A<+\infty$ is constant and $\rho\bigl(f'(x)^{-1}\bigr)$ denotes the spectral radius of $f'(x)^{-1}$.

Chamberland's conjecture. The mapping $f$ is injective.

This is weaker than Alexandrov's conjecture because it does not assert convexity of the image. The source states that it remains open, while related injectivity theorems hold under stronger conditions.