= Solution
The two crossings of $j,h$ make their full <anyon monodromy>. For a <transparent anyon> $h$, this is the identity, so that pair of crossings can be removed together. No single-crossing exchange phase, and no <topological spin> from a twist, is introduced.
After this removal the upper and lower <anyon fusion trees> differ only in their intermediate labels. In the <physics normalization of anyon diagrams>, successive split-fuse bubbles give the <Kronecker deltas> $\delta_{bf}$ and $\delta_{cg}$. The remaining closed normalized state has norm one. Hence, for the admissible labelings of <1/a>,
$$
\boxed{\mathcal A_{bc;fg}=\delta_{bf}\delta_{cg}.}
$$
In particular, every admissible diagram satisfying the label matching in <1/a> evaluates to $1$. There is no factor $d_a$ here: the source explicitly permits the <physics normalization of anyon diagrams>, in which a closed normalized state has value $1$.
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