Let $W$ be a standard $d$-dimensional Brownian motion.

Suppose $b: \mathbb R^d \to \mathbb R^d$ is measurable and bounded. Consider, for every $\varepsilon > 0$, the solution $X^\varepsilon$ on $[0, T]$ to the SDE

$$dX^\varepsilon_t = b(X_t) \, dt + \varepsilon \, dW_t, \, X_0 = 0$$

It is known that the SDE admits weak solutions for every $\varepsilon$.

**Question:** Do the solutions $X^\varepsilon_t$ converge in law in $C[0, T]$ to some (possibly nondeterministic) limit as $\varepsilon \to 0$?