Density conjecture for 2-paths among piecewise Hölder-differentiable curves

Let $C^{1,\beta}$ denote the space of differentiable functions $\gamma:[0,1]\to {\mathbb{R}}^2$ with derivative $\dot \gamma\in C^\beta$, where $\beta\in(0,1]$. For a curve $P$, let $S(P,r)$ be the enclosed area associated with radius $r$, and call $P$ a 2-path if there is an $r$ such that $S(P,r)=\pi r^2$.

Density conjecture. The subset of piecewise $C^{1,\beta}$ 2-paths is dense in $C^{1,\beta}$.

This conjecture formalizes the empirical observation that, for most paths, two copies of the path admit a radius producing a trajectoid. The statement concerns density in an infinite-dimensional function space, where genericity and measure-theoretic claims are delicate; no resolution is given here.