= Solution
The <Gronwall inequality> applied to part (a) gives
$$
\boxed{\mathbb E f_\epsilon(Z_{t\wedge\tau})\leq\frac{e^{(L+K^2)t}}{z_0+\epsilon}.}
$$
Continuity implies $Z_\tau=0$ on $\{\tau<\infty\}$. Thus on $\{\tau\leq t\}$ the stopped reciprocal equals $1/\epsilon$. Positivity gives
$$
\mathbb P(\tau\leq t)\leq\epsilon\mathbb E f_\epsilon(Z_{t\wedge\tau})\leq\frac\epsilon{z_0+\epsilon}e^{(L+K^2)t}.
$$
Let $\epsilon\downarrow0$, using $z_0>0$, to obtain $\mathbb P(\tau\leq t)=0$. Taking the union over integer $t$ proves $\tau=\infty$ almost surely. Hence the <strict order preservation for scalar Lipschitz diffusions> conclusion is
$$
\boxed{\mathbb P(X_t<Y_t\text{ for every }t\geq0)=1,}
$$
which is stronger than the requested weak inequality. The single-event argument prevents a gap between fixed-time ordering and simultaneous ordering at all times.
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