= Solution
For independent exponential survival times with total observed <person-time at risk> $T_\bullet$ and $d$ events, the maximum-likelihood rate is $\widehat h=d/T_\bullet$. A two-group exponential model is a proportional-hazards model because both hazards are constant, so their ratio is constant.
Here
$$
\widehat h_0=\frac{1+k}{t_1+t_4},
\qquad
\widehat h_1=\frac1{t_2+c}.
$$
Thus the group-1 to group-0 hazard-ratio estimate is
$$
\boxed{\widehat\lambda
=\frac{t_1+t_4}{(1+k)(t_2+c)}}.
$$
It decreases continuously as $c$ increases. The parametric exponential likelihood uses exact exposure times through each arm's person-time, whereas the Cox partial likelihood uses only which subjects belong to each event's risk set. For $t_2<c<t_4$, changing $t_4$ alters group 0 person-time and hence $\widehat\lambda$, but subject 4 is alone if it fails at $t_4$, so that event contributes one to the partial likelihood and $\widehat\theta$ is unchanged.
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