Let $X$ be the diffeological space of germs of paths $c: \mathbb{R} \rightarrow \mathbb{R}^n$, where two paths $c_1, c_2$ are equivalent if $c_1(t) = c_2(t)$ for all $t$ in some interval $(-\varepsilon, \varepsilon)$ (the diffeology on $X$ is the quotient diffeology induced from the diffeological space $C^\infty(\mathbb{R}, \mathbb{R}^n)$).
Q: Are all vector bundles on $X$ trivial?
Clearly, $X$ is smoothly contractible to a point, but that does not necessarily mean that all bundles on $X$ are trivial, see section 3 of this paper. However, this would follow if we knew that all bundles on $X$ are "$D$-numerable".
