Smooth density conjecture for inextensible centerlines with nonvanishing curvature

Let $I$ be the interval under consideration, let $\boldsymbol{A}$ and $\boldsymbol{B}$ be the prescribed endpoint values of $D^{[1]}\mathbf{u}$, and fix $\kappa_m>0$ and $p\geq 4$. Define

$$\Xi^{\kappa_m,p}=\{\mathbf{u}\in W^{3,p}(I;\mathbb{R}^3):D^{[1]}\mathbf{u}(0)=\boldsymbol{A},\ D^{[1]}\mathbf{u}(1)=\boldsymbol{B},\ |\mathbf{u}'(s)|=1\ \text{and}\ |\mathbf{u}”(s)|\geq\kappa_m\ \text{for a.e. }s\in I\}.$$

Here $X^{\kappa_m,p}$ denotes the corresponding space of centerlines used in the paper.

Smooth density conjecture. Smooth functions are strongly dense in $\Xi^{\kappa_m,p}$. In particular,

$$X^{\kappa_m,p}=\Xi^{\kappa_m,p},$$

and consequently $X^{\kappa_m,p}$ is sequentially weakly closed.

This density property would support the direct-method analysis of the elastic energy for inextensible ribbons whose centerlines have curvature bounded away from zero, and would complete the identification of the relevant admissible space with its smooth approximation. The source presents this as a natural conjecture and gives no resolution.