Let $f: [0, 1] \to \mathbb R$ be a bounded, continuous function, and $W$ a standard Brownian motion.

Denote $Y := \int_0^1 f(t) \, dW_t$.

For each $\varepsilon > 0$, consider the conditioned random variable $Y_\varepsilon := \varepsilon Y | \{W_1 \geq \frac{1}{\epsilon}\}.$

Question: Is it true that $Y_\varepsilon$ converges in law to the deterministic random variable $\int_0^1 f(t) \, dt$ as $\varepsilon \to 0$?