Conjecture on the Haagerup graphs $\Gamma_k$ as principal graphs

Let $\Gamma_k$ denote the graphs in the paper's Figure 2. Haagerup graph principal-graph conjecture. For every $k>1$, the graph $\Gamma_k$ is not a principal graph of a subfactor. The paper proves this for $13\geq k>1$ and indicates that the same conclusion is very likely for $19\geq k>13$, conditional on the irreducibility issue; the assertion for all $k>1$ remains open.