= Solution
Let $S_A,S_B$ be the distinct group <survival functions>, and let $H_*$ denote the common pooled <cumulative hazard function> used for the residuals. At a point where its inverse is defined, residual survival in group $g$ is
$$
S_{u,g}(u)=S_g(H_*^{-1}(u)).
$$
Thus the group residual curves generally differ. On intervals where $S_A(t)>S_B(t)$, the same common increasing transformation gives $S_{u,A}(u)>S_{u,B}(u)$: the better-surviving group has more large residual failure times. If the original survival curves cross, the residual curves can also cross, so no fixed ordering or <proportional hazards> assumption is implied.
\b[Distinct group distributions generally produce distinct residual survival curves, even when the pooled residual curve looks exponential.] Separate correctly fitted group hazards would restore the unit-exponential target within each group, but that is a different transformation from the common fit under discussion.
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