= Solution
Use arrangement III. Lines 2 and 4 connect B and C to A, line 3 connects C to B, line 1 connects D to C, and line 5 connects D to A. Every measured multipole preserves <parity>, so \b[B, C and D all have positive <parity>]. The two mixed M1+E2 transitions to $J_A=7/2$ strictly imply
$$
J_B,J_C\in\{5/2,7/2,9/2\}.
$$
The measured E2 C-to-B and D-to-C transitions require $|J_C-J_B|\le2$ and $|J_D-J_C|\le2$, with their angular sums at least two. The M3 D-to-A line gives $|J_D-7/2|\le3$ and $J_D+7/2\ge3$. These inequalities, with half-integer <spins>, are the strict observed-line constraints. They do not alone fix unique <spins>.
The customary low-multipole-dominance inference goes further. If lower-order allowed radiation would not be accidentally hindered, a pure M3 D-to-A decay indicates $|J_D-7/2|=3$, giving $J_D=1/2$ or $13/2$. Its E2 connection to C then forces $J_C=5/2$ or $9/2$, respectively. Treating the pure E2 C-to-B transition similarly as a <spin> change of two selects
$$
\boxed{(J_B^P,J_C^P,J_D^P)
=(\tfrac{9}{2}^{+},\tfrac{5}{2}^{+},\tfrac{1}{2}^{+})
\quad\text{or}\quad(\tfrac{5}{2}^{+},\tfrac{9}{2}^{+},\tfrac{13}{2}^{+}).}
$$
In either assignment the missing D-to-B line changes <spin> by four. Its lowest parity-preserving multipole is E4, so it is naturally much weaker than the observed E2 D-to-C branch. This spin-reflection ambiguity around the reference <spin> is not resolved by the spectra under this approximation.
The boxed alternatives are \b[conditional on ordinary, unhindered lower-multipole strengths]. If accidental or structural hindrance is allowed, other <spins> satisfying the preceding inequalities remain possible: an upper limit on an unobserved lower multipole does not mathematically prove its selection-rule prohibition. The supplied <intensities> and absence of other comparable lines support the usual assignments but do not supply measured reduced strengths sufficient to eliminate that qualification.
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