The irreducible subfactor $(R^{S_3} \subset R)$, of index $6$, admits a principal graph with a multiplicity $2$ edge because the group $S_3$ admits an irreducible complex representation of dimension $2$.
The same for the subfactor $(R \subset M_2(R))$, of index $4$, but it is not irreducible.
Question: Is $6$ the smallest index for an irreducible subfactor to have a principal graph with an edge of multiplicity $>1$?