Let $X$ be the solution to the one dimensional SDE

$dX_t = \mu(t, X_t)dt + \sigma(t, X_t) dW_t$, for $t \in [0, T]$.

with $X_0= x_0$ a.s. for some $x_0 \in \mathbb R$.

Here $W_t$ denotes a standard Brownian motion, and we assume $\mu$ and $\sigma$ are Lipschitz continuous and uniformly bounded.

For every $\varepsilon > 0$, denote by $\mathcal S_{\varepsilon}$ the event $\sup_{t \in [0, T]} |W_t| \leq \varepsilon$.

**Question:** Considering $X$ as a $C[0, T]$-valued random variable, is it true that the conditioned random variables $X| \mathcal S_\varepsilon$ converge in law to the deterministic solution $Y_t$ of

$dY_t = \mu(t, Y_t) dt$, with $Y_0 = x_0$ a.s.?