Assume we have a one dimensional rough path $\mathbf{X} = (X, \mathbb{X})$ with finite $p$-variation, with $p \in (1,2]$. Assuming that $Y$ is controlled in the sense of Gubinelli with respect to the path $X$. Then, what is the $p$-variation of the rough path integral $$F(x) = \int_0^x Y \ d\mathbf{X}$$ By known estimates (see chapter IV of introduction to rough path by Hairer and Fritz),
$$ |F(t) - F(s)| \le Y_s X_{s,t} + Y_s'\mathbb{X}_{s,t} + C|t-s|^{3/p}$$ for some constant $C$ depending on the $p$ and $p/2$ variation of the path and gubinelli's derivative/residue.
Hence, it seems that it's just of finite $p$-variation. But shouldn't it gain more regularity as we are considering the integral, i.e. getting to $p/2$ finite variation?
The estimate I see in 4.10 thm in Hairer-Friz i s a bit better
$$|\int_{s}^{t}Y_{r}dX_{r}-Y_s X_{s,t} + Y_s'\mathbb{X}_{s,t}|\leq C|t-s|^{3\alpha}.$$
But yes the best possible is indeed p-variation because after all the Gubinelli derivative is defined as
$$\int_{s}^{t}Y_{r}dX_{r}=\lim_{|P|\to 0}\sum_{s,t\in P}Y_s X_{s,t} + Y_{s}' \mathbb{X}_{s,t}$$
and so we have to deal with the restricted regularity of $X_{s,t}$ at the very least. This situation is similar to that of Ito integrals, where at best we have $(\frac{1}{2})^{-}-$Hölder regularity.