= Solution
Assume $d<c$. Directly comparing the two piecewise densities gives
$$
\frac{g_1(x)}{g_0(x)}=
\begin{cases}
d,&r(x)\leq d,\\
r(x),&d<r(x)<c,\\
c,&r(x)\geq c.
\end{cases}
$$
Since
$$
\log r(x)=\Delta\left(x-\frac{\theta_0+\theta_1}{2}\right),
$$
we obtain
$$
\log\frac{g_1(x)}{g_0(x)}
=\left[\Delta\left(x-\frac{\theta_0+\theta_1}{2}\right)\right]_{\log d}^{\log c}.
$$
Thus the sample log-likelihood ratio is $nS_n$ with truncation values $a=\log d$ and $b=\log c$. Rejecting for large $S_n$ is exactly the likelihood-ratio test between the two least-favorable contaminated distributions.
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