= Solution
Read the arrows upwards through the central part of the diagram. The lower <anyon fusion tree> first splits $a$ into $b,c$, then $b$ into $d,j$ and $c$ into $h,e$. The upper <anyon fusion tree> combines $d,j$ into $f$, $h,e$ into $g$, and finally $f,g$ into $a$. The two interior crossings are not vertices. Thus admissibility requires the six <anyon fusion multiplicities>
$$
N^a_{bc}N^b_{dj}N^c_{he}N^f_{dj}N^g_{he}N^a_{fg}=1.
$$
If any factor vanishes, the diagram has zero value.
Make an <anyon vertex gauge transformation> by multiplying each splitting vertex $z\to x,y$ by an independent <wave phase> $u^{xy}_z$ and its adjoint fusion vertex by $(u^{xy}_z)^{-1}$. The diagram changes by
$$
G=\frac{u^{bc}_a u^{dj}_b u^{he}_c}{u^{fg}_a u^{dj}_f u^{he}_g}.
$$
Therefore the label-matching condition for cancellation of all <anyon vertex gauge transformation> factors is
$$
\boxed{b=f,\qquad c=g,}
$$
with $a,d,e,h,j$ otherwise unrestricted apart from the <anyon fusion rules>. This statement also allows coincident labels: the three ordered vertex triples in the numerator must equal those in the denominator as multisets; matching the triples in any of the six possible orders still forces $b=f,c=g$. Conversely, these equalities visibly give $G=1$.
Closure of the drawing alone does not guarantee invariance under an <anyon vertex gauge transformation>: unmatched fusion and splitting vertices retain their independently chosen basis phases. A zero value is, of course, invariant regardless of phases; the boxed condition describes cancellation of the vertex factors themselves.
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