= Solution
Integrating the stopped <stochastic differential equation> for $Y$ and then taking a common full-probability event for a <localizing sequence> gives, simultaneously for every $t<T$,
$$
Y_t=\sqrt{x_0}+\frac12B_t-\frac18\int_0^t\frac{ds}{Y_s}.
$$
The integral is finite on each compact time interval before $T$: the continuous positive path has a positive minimum there. Its integrand is nonnegative, so
$$
\boxed{W_t-Y_t=\frac18\int_0^tY_s^{-1}\,ds\geq0\quad\text{for every }0\leq t<T\text{ almost surely}.}
$$
This is a pathwise comparison using the same <Brownian motion>, not a comparison of independent distributions. Continuity lets the identities established initially at rational stopped times hold on the entire <stochastic interval>.
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