Let $X^n$ be a collection of smooth functions so that their $\alpha$-Holder norms for $\alpha \in (1/3,1/2)$ are uniformly bounded - that is $\sup_n \|X^n\|_\alpha<\infty$. Define the standard Riemann-Stieltjes integrals $\mathbb X_{s,t}^n=\int_s^t (X^n(r)-X^n(s) )\otimes dX^n(r)$. Then is it true that $\sup_n\|\mathbb X^n\|_{2\alpha}<\infty$?

This is related to rough paths theory and the interpolation result of Friz and Hairer's exercise 2.9. I am wondering if the second condition really needs to be checked.