In [Friz and Hairer's notes on rough paths][1], there is exercise 2.9 which is called the "interpolation theorem". It says that if you have a sequence of rough paths $\mathbf X^n=(X^n,\mathbb X^n)\in \mathscr C^\beta$ with $\beta\in(1/3,1/2)$ so that $\sup_n \|X^n\|_\beta<\infty$ and $\sup_n \|\mathbb X^n\|_{2\beta}<\infty$ with $\mathbf X^n\to \mathbf X$ pointwise, then $\mathbf X\in \mathscr C^\beta$ and $\mathbf X^n\to \mathbf X$ in $\mathscr C^\alpha$ for all $1/3<\alpha<\beta<1/2$. 

Can this be generalized to general rough paths not restricting to $(1/3,1/2)$? 








  [1]: https://www.hairer.org/notes/RoughPaths.pdf