Q-system Completeness of Unitary Connections

A Q-system is a unitary version of a separable Frobenius algebra object in a C*-tensor category. In a recent joint work with P. Das, S. Ghosh and C. Jones, the author has categorified Bratteli diagrams and unitary connections by building a $2$-category $\mathbf{\text{UC}}$. We prove that every $Q$-system in $\mathbf{\text{UC}}$ splits.