I am trying to find a proof or counterexample for this statement regarding the running maximum of two related processes:
Proposition: Given two Ornstein-Uhlenbeck-like processes with $\sigma, \mu > 0$
\begin{align} \mathrm{d} X_t^1 & = \theta \left( \mu - X_t^1 \right) \mathrm{d} t + Y_t \mathrm{d} B_t^1, \\ \mathrm{d} X_t^2 & = \theta \left( \mu - X_t^2 \right) \mathrm{d} t + \sigma \mathrm{d} B_t^2. \label{eq:sigma_process} \end{align} with the same initial condition $x$ and with $|Y_t| \leq \sigma$ for all $t \geq 0$, is it true that for $\lambda > 0$
$$ \mathbb{P}_x \left[ \sup_{0 \leq t \leq T} X_t^1 \geq \mu + \lambda \right] \leq \mathbb{P}_x \left[ \sup_{0 \leq t \leq T} X_t^2 \geq \mu + \lambda \right]? $$
Intuitively it seems like this should be true, but I have not found a way to prove it. Nor have I found any counter-examples when experimenting with numerical simulations. The related results I have found are various Martingale inequalities and Sleipan's lemma for Gaussian processes, but I have not been able to transfer the techniques used to prove those results to this case.
A similar question was asked here, but this one is different in that the drift dynamics are decreasing in the region $x \geq \mu$, so only the stochastic part of the dynamics can push the state further into that region. Therefore it seems reasonable that the probability should be monotonic in the volatility magnitude.