$C^0$-approximation by smooth curves with prescribed curvature

Question

A theorem by Ghomi (https://link.springer.com/article/10.1007/s10711-007-9151-y) says that any $C^2$ curve $\gamma:[a,b]\to\mathbb R^{n≥3}$ can be $C^1$-approximated by curves with prescribed constant curvature as long as the prescribed curvature is larger than the maximum of the curvature of $\gamma$. I wonder if one could relax the assumption on the prescribed curvature to be constant and weaken the $C^1$-approximability by $C^0$-approximability. More precisely I wonder if the following statement holds true:

Let $\gamma:[a,b]\to\mathbb R^{n≥3}$ be a curve of class $C^2$ with curvature $k$ and let $k':[a,b]\to\mathbb R$ be a smooth function that satisfies $k'>k$ point wise. There is a smooth curve $\delta:[a,b]\to\mathbb R^{n≥3}$ with curvature $k'$ that is $C^0$-close to $\gamma$.

I'd be thankful for any reference.

EDIT: I corrected the question, thanks.

Answer 1

This has been established in a recent paper by Micha Wasem:

h-principle for curves with prescribed curvature. Geom. Dedicata 184 (2016), 135–142.

In particular Wasem showed that one can prescribe curvature as you suggest, and it is not even necessary to give up $C^1$ closeness.