Nonvanishing Euler characteristic for irreducible subfactor planar algebras

Let $\mathcal{P}$ be an irreducible subfactor planar algebra. Define its Euler characteristic by

$$\chi(\mathcal{P}):= -\sum_{b \in [e_1,\operatorname{id}]} \mu(b,\operatorname{id})|\operatorname{id}:b|.$$

Nonvanishing Euler-characteristic conjecture. The Euler characteristic $\chi(\mathcal{P})$ is nonzero.

This conjecture generalizes K.S. Brown's problem from finite groups to irreducible subfactor planar algebras. Gaschütz proved the usual Brown problem for solvable groups; the general planar-algebraic statement remains open.