I am wondering that the following statement is true or not:

Let $(M,g)$ be a complete non-compact Riemannian manifold with $0 < Sect \leq C\cdot dist(O,x)^{-2a}$, $a\in(0,1]$. ($O$ is a point in $M$ and $Sect$ is the sectional curvature.) Then the injectivity radius $inj(x)$ has a lower bound $c\cdot dist(O,x)^{a}$ for some constant $c>0$.

I believe this is true at least for simply-connected manifolds, but I cannot prove it even when $M$ is diffeomorphic to $\mathbb{R}^n$.

Thanks for any comments.

Edit: According to Igor's comment and answer, this question is only undetermined for $n\geq 3$.