We will follow the Lemma 8.10. (Rough Gronwall) and Proposition 8.12 from "a course in rough paths" but modify them for this particular setting of studying
$$Y_{t}=Y_{0}+\int^{t}_{0} Y dX,$$
for $Y\in C^{\alpha},X\in C^{\beta}$ where $\alpha+\beta>1$ but $\beta<\frac{1}{2}$ and $\alpha>\frac{1}{2}$.
The Lemma 8.10. (Rough Gronwall) comes verbatim with no changes
Assume $Y\in C([0,1])$, $r<1$ and $$|Y_{s,t}|\leq M ||Y||_{\infty;[s,t]}|t-s|^{r},(*)$$ for $0\leq s< t\leq 1$. Then there exists $c=c_{r}<\infty$ such that $$||Y||_{\infty;[0,1]}\leq cexp(cM^{1/r})|Y_{0}|.$$
So it suffices to prove (*). From a slight variation of the proof of Proposition 8.12 we have the bound
$$|Y_{s,t}|=|\int_{s}^{t}Y_{r}dX_{r}|\leq c||Y||_{\infty;[s,t]}||X||_{\beta;[s,t]}|t-s|^{\beta}\leq c||Y||_{\infty;[s,t]}|t-s|^{\beta},$$ where we used that $\beta<\frac{1}{2}\leq \alpha$ and so $||Y||_{\alpha;[s,t]}\leq ||Y||_{\beta;[s,t]}$. So the result follows from here.