We define the $\alpha$-Hölder semi-norm as $$\| X\|_\alpha = \sup_{s\neq t \in [0,T]} \frac{|X_{s} - X_t|}{|s-t|^\alpha}$$ and the distance function between two rough paths as, $$\rho_{\alpha}(\mathbf{X}, \tilde{\mathbf{X}}) = \| X - \tilde{X}\|_\alpha + \| \mathbb{X} - \tilde{\mathbb{X}} \|_{2\alpha},$$ where $\mathbb{X}$ is the enhanced path.
Denoting $\mathbf{B}^\text{ito}$ as the enhanced Brownian motion with respect to the Ito integration, i.e. we denote with $B_\cdot(\omega)$ a brownian motion, whose path is determined by a paramter $\omega$, then $$ \mathbf{B}^{\text{ito}}_{s}(\omega) = \left(B_s(\omega), \mathbb{B}^{\text{ito}}_{s} := \int_0^s B_{r}(\omega) dB_r(\omega)\right).$$
Then, given two distinct paths $\mathbf{B}^{\text{ito}}_{s}(\omega)$ and $\mathbf{B}^{\text{ito}}_{s}(\tilde{\omega})$ for $\omega \neq \tilde{\omega}$, what is a good upper bound for $$\rho_{\alpha}(\mathbf{B}^{\text{ito}}_{s}(\omega), \mathbf{B}^{\text{ito}}_{s}(\tilde{\omega})) \le ?$$
My issue is that any estimate I can think of are in probability. So, at best I can think of taking $\omega$ such that it maximizes the path and $\tilde{\omega}$ which minimizes the path, so to have the largest increments. But even in that case we'd have something that doesn't seem bounded (at least it is almost surely, but not for any path, right?) or am I wrong? Any hint is appreciated!