= Solution
Apply the final splitter only to the first ket. The $|dd\rangle$ amplitudes from $i|uv\rangle$ and $|vv\rangle$ cancel, giving
$$
\boxed{|\Psi_{IR}\rangle=-\frac12|\gamma\rangle+\frac1{2\sqrt2}\left(-|cc\rangle+2i|cd\rangle+i|dc\rangle\right).}
$$
Thus detection at $D^+$ occurs only in the coincidence $D^+,C^-$, with <probability> $1/8$. The other two nonzero coincidence probabilities are $P(C^+,C^-)=1/8$ and $P(C^+,D^-)=1/2$. The cancellation is quantum <interference> between indistinguishable surviving alternatives.
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